competitive swarm optimizer (CSO) is considered as a prominent paradigm for solving large-scale multiobjective optimization problems (LMOPs). However, the pairwise random competition (PRC) mechanism used in most existing CSOs may limit their performance in solving LMOPs due to the following reasons. First, when the winner particle obtained by PRC is of poor quality, it may limit the learning effect of its corresponding loser particle. Second, due to the stochastic nature of PRC, the evolutionary direction of the loser particles may be drastically perturbed over the iterations, thus slowing down their convergence speed. To alleviate the above issues, this article proposes a constrained learning (CL)-based CSO for tackling LMOPs, called CL-CSO. First, CL-CSO adopts a set of reference vectors to divide the original objective space into several subregions. Second, CL-CSO designs a CL-based strategy, including the intra-subregion learning and cross-subregion learning strategy, which let the loser particles only learn from the winner particles in their intra-subregions or neighboring subregions, respectively. Moreover, CL-CSO designs a Gaussian model assisted evolutionary strategy to help the evolution of winner particles, aiming to further improve the diversity and quality of winner particles. This way, the learning effect of particles and the overall convergence speed can be significantly enhanced. Compared to several competitive algorithms for tackling LMOPs, experimental results show that CL-CSO performs well in solving two well-known benchmark LMOPs (containing 2-3 objectives and 500-5000 decision variables), as well as real-world instance selection problems.

Machine learning-based optimization frameworks have attracted increasing attention for accelerating the solution of large-scale quadratically constrained quadratic programs (QCQPs) by exploiting shared problem structure across instances. However, existing machine learning (ML) frameworks often rely on the assumption of parametric models and large-scale solvers. This article introduces HEQP, a hypergraph neural network-based evolutionary optimization framework for large-scale QCQPs. This framework features two main components: 1) hypergraph-based neural prediction, which predicts optimal solutions for QCQPs without assumptions of models; and 2) evolutionary large neighborhood search (Evo-LNS), which employs a McCormick relaxation-based repair strategy to search and apply crossover on neighborhood solutions using a small-scale solver. We further show that our framework is equivalent to the interior-point method (IPM), a polynomial-time algorithm, for quadratic programming. Experiments on two types of benchmark problems and 13 large-scale real-world instances from the QPLIB illustrate that our framework outperforms state-of-the-art solvers (including Gurobi, SCIP, and SHOT) in both solution quality and time efficiency, highlighting the efficiency of ML-based optimization frameworks for QCQPs.

The Nearest-Better Network (NBN) is a powerful method to visualize sampled data for continuous optimization problems while preserving multiple landscape features. However, the calculation of NBN is very time-consuming, and the extension of the method to combinatorial optimization problems is challenging but very important for analyzing the algorithm’s behavior. This paper provides a straightforward theoretical derivation showing that the NBN network essentially functions as the maximum probability transition network for algorithms. This paper also presents an efficient NBN computation method with logarithmic linear time complexity to address the time-consuming issue. By applying this efficient NBN algorithm to the OneMax problem and the Traveling Salesman Problem (TSP), we have made several remarkable discoveries for the first time: The fitness landscape of OneMax exhibits neutrality, ruggedness, and modality features. The primary challenges of TSP problems are ruggedness, modality, and deception. Three state-of-the-art TSP algorithms (EAX, LKH, and NLKH) have limitations when addressing challenges related to modality and deception, respectively. LKH, based on local search operators, fails when there are deceptive solutions near global optima. EAX, which is based on a single population, can efficiently maintain diversity. However, when multiple attraction basins exist, EAX retains individuals within multiple basins simultaneously, reducing inter-basin interaction efficiency and leading to algorithm’s stagnation. NLKH improves over LKH by leveraging learned edge weights to increase the chance of reaching the global basin, but it remains vulnerable to deceptive funnels due to biased learning from underrepresented complex instances.

Machine Learning (ML)-based optimization frameworks emerge as a promising technique for solving large-scale Mixed Integer Linear Programs (MILPs), as they can capture the mapping between problem structures and optimal solutions to expedite their solution process. However, existing solution frameworks often suffer from high model computation costs, incomplete problem reduction, and reliance on large-scale solvers, leading to performance bottlenecks in ultra-large-scale problems with complex constraints. To address these issues, this paper proposes Light-EvoOPT, a Lightweight Evolutionary Optimization Framework for Ultra-Large-Scale Mixed Integer Linear Programs, which can be divided into four stages: (1) Problem Formulation for problem division to reduce model computational costs, (2) Model-based Initial Solution Prediction for predicting and constructing the initial solution using a small-scale training dataset, (3) Problem Reduction for both variable and constraint reduction, and (4) Evolutionary Optimization for current solution improvement employing a lightweight optimizer. Experiments on four benchmark datasets with tens of millions of variables and constraints and a real-world problem show that the proposed framework based on the sole use of a lightweight optimizer, trained on only one-thousandth of the scale of ultra-large-scale problems, is able to outperform state-of-the-art ML-based frameworks and advanced solvers (e.g. Gurobi) within a specified computational time, validating the feasibility and effectiveness of our proposed ML-based evolutionary optimization framework for ultra-large-scale MILPs.

Code smells signal poor software design that can prevent maintainability and scalability. Identifying code smells is difficult because of the large volume of code, considerable detection expenses, and the substantial effort needed for manual tagging. Although current techniques perform well in within-project situations, they frequently struggle to adapt to cross-project environments that have varying data distributions. In this paper, we introduce CLADES (Cross-project Learning and Adaptation for Detection of Code Smells), a hybrid evolutionary approach consisting of three main modules: Initialization, Evolution, and Adaptation. The first module generates an initial population of decision tree detectors using labeled within-project data and evaluates their quality through fitness functions based on structural code metrics. The evolution module applies genetic operators (selection, crossover, and mutation) to create new offspring solutions. To handle cross-project scenarios, the adaptation module employs a clustering-based instance selection technique that identifies representative instances from new projects, which are added to the dataset and used to repair the decision trees through simulated annealing. These locally refined decision trees are then evolved using a genetic algorithm, thus enabling continuous adaptation to new project instances. The resulting optimized decision tree detectors are then employed to predict labels for the new unlabeled project instances. We assess CLADES across five open-source projects and we show that it has a better performance with respect to baseline techniques in terms of weighted F1-score and AUC-PR metrics. These results emphasize its capacity to effectively adjust to different project environments, facilitating precise and scalable detection of code smells while minimizing the need for manual review, contributing to more robust and maintainable software systems.

The development of specialized, domain-specific operators has significantly enhanced the performance of evolutionary algorithms for solving optimization problems. However, creating such operators often requires substantial effort from human experts, making the process slow, resource-intensive, and heavily reliant on domain knowledge. To overcome these limitations, generation hyper-heuristics provide a framework for automating the design of variation operators by evolving combinations of heuristic components without direct expert input. In this work, we propose a generation hyper-heuristic method based on grammatical evolution to automatically design variation operators (crossover and mutation) tailored to the multi-objective quadratic assignment problem (mQAP)—a challenging combinatorial optimization problem with many real-world applications. Using the proposed method, variation operators were generated considering six mQAP instances with two and three objectives, leveraging MOEA/D as a multi-objective optimizer. For validation, the generated operators were evaluated on unseen instances. Our experimental results indicate that the evolved operators enhance the performance of MOEA/D compared to standard crossover operators. Furthermore, the top-performing operator in training did not always generalize best to larger instances, while some lower-ranked operators showed better adaptability. These results highlight the potential of automated operator design in effectively tackling complex optimization problems like the mQAP.

The hypervolume (HV) indicator is widely used in multi-objective evolutionary algorithms (MOEAs) due to its Pareto compliance property. This property makes it very effective for assessing the quality of approximation sets from different MOEAs and for ranking solutions among a population of solutions using the individual hypervolume contribution (HVC). However, the computational cost of computing the HV increases exponentially with the number of objectives. Furthermore, this cost increase is worse when the HVC is adopted as a density estimator, making it prohibitive in many-objective optimization problems (MaOPs). In this work, we propose a novel approach to create HVC approximation functions using Grammatical Evolution (GE). We describe the grammar and fitness functions designed to identify the worst-contributing individual given a population of non-dominated solutions. Then, we use a GE implementation with training data generated from the DTLZ and WFG test problems. The resulting approximation functions, tailored for dimensionalities ranging from 2 to 10, are evaluated against two state-of-the-art methods: HVC-Net and the R2-based HVC approximation. Experimental results on validation data also derived from benchmark problems show that our GE-generated functions consistently outperform both alternative approaches regarding worst-contributing individual identification for dimensions greater than two, while maintaining competitive execution times. These results indicate that GE is a viable and effective tool for generating high-quality HVC approximations, particularly suitable for solving MaOPs.

Network Intrusion Detection System (NIDS) has become more important as a large number of diverse devices connect to the Internet of Things (IoT). Generally, training an effective NIDS requires a large amount of high-quality and centralized attack data. However, in real-world scenarios, it is difficult to centralize the distributed data for training NIDS in the IoT due to the privacy concerns and data format heterogeneity. To solve this problem, a novel NIDS combining federated learning and cost-sensitive learning is proposed, named FIDS-CL. Specifically, federated learning with dynamic weights aggregation tackles the problem of non-clusterable data, where multiple clients collaboratively enhance the overall performance while dynamically aggregating weights to maximize the retention of high-performing client models. Moreover, cost-sensitive learning is employed to alleviate the problem of class imbalance in NIDS by dynamically adjusting the gradient descent weights of different classes in the loss function, thereby emphasizing the importance of minority classes. Therefore, our method can effectively handle data imbalance while safeguarding data privacy of clients, which is more effective to detect network intrusions. The experiments conducted across various scenarios validate the superior detection capabilities and computational efficiency of FIDS-CL when compared to other state-of-the-art NIDSs.

Fitness landscape analysis (FLA) is quite important in evolutionary computation. In this article, we propose a novel FLA method, the nearest-better network (NBN), which uses the nearest-better relationship to simplify the original fitness landscape of continuous optimization problems. We introduce an efficient algorithm to calculate NBN for continuous problems. We also propose four numerical measurements and a 3-D visualization method based on NBN. Experiments show that compared to the other main FLA methods, the four numerical measurements proposed here can effectively measure the four intended features: 1) neutrality; 2) ruggedness; 3) modality; and 4) Basin of Attraction, respectively, and common features of the fitness landscape can be maintained in 3-D NBN visualization, regardless of the scale of the problem. NBN also provides a view of how algorithms search in high-dimensional problems with the help of the 3-D NBN visualization.

Feature selection (FS) is a very important technique for hyperspectral image (HSI) classification, as successfully selecting informative features can significantly increase the learning performance while reducing the computational cost. However, most of the existing FS methods tend to treat the HSI as a whole for FS, which does not fully consider the unique characteristics of HSIs and disregards the fact that different feature classes possess varying preferences for features. Thus, this paper proposes a superpixel segmentation based evolutionary multitasking algorithm for FS of HSIs, called SS-EMT. First, the superpixel segmentation method is used to partition the original HSI into several superpixel blocks, which can preserve well the information of different classes of the original image. Second, in order to explore each superpixel block efficiently, an evolutionary multitasking algorithm using particle swarm optimization is designed, which treats each superpixel block as a subtask and then optimizes these subtasks collaboratively by transferring useful knowledge among related subtasks. In addition, a new individual evaluation mechanism is devised to obtain multiple high-quality feature subsets with different numbers of features simultaneously in a single run, thus reducing the computational cost. Finally, extensive experimental results on four common HSI datasets under three classifiers validate that our proposed method outperforms several state-of-the-art FS methods.

This paper presents a survey of applications of multi-objective evolutionary algorithms in several biotechnology areas. The application areas covered in the survey include: molecular docking, metabolic engineering, synthetic biology, optimization of industrial bio-processes and data processing for bioinformatics (which covers multiple sequence alignment and feature selection and classification for diagnosis of diseases). In the final part of the paper, some potential areas for future research are briefly discussed.

In recent years, evolutionary computation has signif-icantly advanced in processes related to machine learning. How-ever, the reciprocal integration of machine learning techniques into evolutionary computation remains relatively unexplored. Machine learning can substantially enhance the understanding of processes within Multi-Objective Evolutionary Algorithms (MOEAs) by harnessing its proficiency in identifying patterns and employing data-driven approaches. Existing studies lack a comprehensive understanding of the intricate interaction between machine learning models and evolutionary algorithms, necessi-tating prioritized investigation to ensure the efficacy, reliabil-ity, and compatibility of integrated models within optimization frameworks. This paper addresses this gap by examining the behavior of using Generative Artificial Intelligence (AI) models as a population variation operator in Multi-objective Optimization Problems. Our experimental results reveal that Generative AI, particularly Distributional Adversarial Networks (DANs), sur-passes the performance of a traditional Generative Adversarial Network. Furthermore, DANs improve the population by gen-erating novel non-dominated solutions and augmenting overall performance and diversity. This study reveals the potential of the integration of Generative AI in evolutionary computation, presenting a pathway for advancements in addressing common challenges within multi-objective optimization problems.

The Team Formation Problem in Social Networks (TFP-SN) describes the process of finding an effective group of people, drawn from a network of experts, to perform a particular task. For a team to be considered as effective, it requires to comply with a task-specific skills set while also showing a high degree of cohesiveness. Although team effectiveness is subject to multiple criteria, the study of the problem from a multi-objective (MO) perspective is still scarce. In this paper, we focus on an MO TFP-SN whose objective is to maximize the team's level of expertise and the team's density, simultaneously. To solve this problem, we introduce two novel recombination operators to be used within the framework of the well-known NSGA-II. Our proposed crossover operators act as heuristics that compute the parents' unique and shared information, which is then combined for generating potentially improved offspring. Our experiments show that each of the two proposed crossover operators lead to significantly better results when compared to a naive crossover operator taken from the specialized literature. Particularly, the results consistently show higher hypervolume values when compared to the use of an adaptation of a simple recombination operator commonly used for this problem. The good performance of our proposed operators may be attributed to the incorporation of knowledge that exploits the structure of the problem.

Supervised machine learning techniques include classification and regression. In regression, the objective is to map a real-valued output to a set of input features. The main challenge that existing methods for regression encounter is how to maintain an accuracy-simplicity balance. Since Regression Trees (RTs) are simple to interpret, many existing works have focused on proposing RT and Model Tree (MT) induction algorithms. MTs are RTs with a linear function at the leaf nodes rather than a numerical value are able to describe the relationship between the inputs and the output. Traditional RT induction algorithms are based on a top-down strategy which often leads to a local optimal solution. Other global approaches based on Evolutionary Algorithms (EAs) have been proposed to induce RTs but they can require an important calculation time which may affect the convergence of the algorithm to the solution. In this paper, we introduce a novel approach called Bi-level Evolutionary Model Tree Induction algorithm for regression, that we call BEMTI, and which is able to induce an MT in a bi-level design using an EA. The upper-level evolves a set of MTs using genetic operators while the lower-level optimizes the Linear Models (LMs) at the leaf nodes of each MT in order to fairly and precisely compute their fitness and obtain the optimal MT. The experimental study confirms the outperformance of our BEMTI compared to six existing tree induction algorithms on nineteen datasets.

The competitive swarm optimizer (CSO) classifies swarm particles into loser and winner particles and then uses the winner particles to efficiently guide the search of the loser particles. This approach has very promising performance in solving large-scale multiobjective optimization problems (LMOPs). However, most studies of CSOs ignore the evolution of the winner particles, although their quality is very important for the final optimization performance. Aiming to fill this research gap, this article proposes a new neural net-enhanced CSO for solving LMOPs, called NN-CSO, which not only guides the loser particles via the original CSO strategy, but also applies our trained neural network (NN) model to evolve winner particles. First, the swarm particles are classified into winner and loser particles by the pairwise competition. Then, the loser particles and winner particles are, respectively, treated as the input and desired output to train the NN model, which tries to learn promising evolutionary dynamics by driving the loser particles toward the winners. Finally, when model training is complete, the winner particles are evolved by the well-trained NN model, while the loser particles are still guided by the winner particles to maintain the search pattern of CSOs. To evaluate the performance of our designed NN-CSO, several LMOPs with up to ten objectives and 1000 decision variables are adopted, and the experimental results show that our designed NN model can significantly improve the performance of CSOs and shows some advantages over several state-of-the-art large-scale multiobjective evolutionary algorithms as well as over model-based evolutionary algorithms.

Multiobjective multitasking optimization (MTO) needs to solve a set of multiobjective optimization problems simultaneously, and tries to speed up their solution by transferring useful search experiences across tasks. However, the quality of transfer solutions will significantly impact the transfer effect, which may even deteriorate the optimization performance with an improper selection of transfer solutions. To alleviate this issue, this article suggests a new multiobjective multitasking evolutionary algorithm (MMTEA) with decomposition-based transfer selection, called MMTEA-DTS. In this algorithm, all tasks are first decomposed into a set of subproblems, and then the transfer potential of each solution can be quantified based on the performance improvement ratio of its associated subproblem. Only high-potential solutions are selected to promote knowledge transfer. Moreover, to diversify the transfer of search experiences, a hybrid transfer evolution method is designed in this article. In this way, more diverse search experiences are transferred from high-potential solutions across different tasks to speed up their convergence. Three well-known benchmark suites suggested in the competition of evolutionary MTO and one real-world problem suite are used to verify the effectiveness of MMTEA-DTS. The experiments validate its advantages in solving most of the test problems when compared to five recently proposed MMTEAs.

Routing and scheduling problems with increasingly realistic modeling approaches often entail the consideration of multiple objectives, time constraints, and modeling the system as a multigraph. This detailed modeling approach has increased computational complexity and may also lead to violation of the additivity property of the costs. In the worst scenario, increased complexity makes the problem intractable for exact algorithms. Even when the problem is solvable, exact algorithms may not provide solutions within the given time budget, and the found solutions are not guaranteed to be optimal due to the additivity property violation. Approximate solution methods become more suitable in this case. This article focuses on one particular real-world application, the Airport Ground Movement Problem, where both time constraints and parallel arcs are involved. We introduce a novel memetic algorithm for routing in multigraphs with time constraints (MARMT) and present a comprehensive study of its different variants based on diverse genetic representation methods. We propose a local search operator that enhances search efficiency and effectiveness. MARMT is tested on real data based on two airports of different sizes. Our results show that MARMT does not suffer from the nonadditivity property problem as it outperforms the state-of-the-art exact algorithm when allowed to converge. When a time budget of 10 s is imposed on MARMT, it is able to provide solutions with quality comparable (within 1%–5% degradation) to the ones given by the exact algorithm with respect to the aggregated objective values. MARMT can be adapted for other applications, such as train operations.

In recent years, surrogate-assisted evolutionary algorithms (SAEAs) have been sufficiently studied for tackling computationally expensive multiobjective optimization problems (EMOPs), as they can quickly estimate the qualities of solutions by using surrogate models to substitute for expensive evaluations. However, most existing SAEAs only show promising performance for solving EMOPs with no more than 10 dimensions, and become less efficient for tackling EMOPs with higher dimensionality. Thus, this article proposes a new SAEA with a simplified helper task for tackling high-dimensional EMOPs. In each generation, one simplified task will be generated artificially by using random dimension reduction on the target task (i.e., the target EMOPs). Then, two surrogate models are trained for the helper task and the target task, respectively. Based on the trained surrogate models, evolutionary multitasking optimization is run to solve these two tasks so that the experiences of solving the helper task can be transferred to speed up the convergence of tackling the target task. Moreover, an effective model management strategy is designed to select new promising samples for training the surrogate models. When compared to five competitive SAEAs on four well-known benchmark suites, the experiments validate the advantages of the proposed algorithm on most test cases.

Feature construction represents a crucial data preprocessing technique in machine learning applications because it ensures the creation of new informative features from the original ones. This fact leads to the improvement of the classification performance and the reduction of the problem dimensionality. Since many feature construction methods require discrete data, it is important to perform discretization in order to transform the constructed features given in continuous values into their corresponding discrete versions. To deal with this situation, the aim of this paper is to jointly perform feature construction and feature discretization in a synchronous manner in order to benefit from the advantages of each process. Thus, we propose here to model the discretization-based feature construction task as a bi-level optimization problem in which the constructed features are evaluated based on their optimized sequence of cut-points. The resulting algorithm is termed Discretization-Based Feature Construction (Bi-DFC) where the proposed model is solved using an improved version of an existing co-evolutionary algorithm, named I-CEMBA that ensures the variation of concatenation trees. Bi-DFC performs the selection of original attributes at the upper level and ensures the creation and the evaluation of constructed features at the upper level based on their optimal corresponding sequence of cut-points. The obtained experimental results on ten high-dimensional datasets illustrate the ability of Bi-DFC in outperforming relevant state-of-the-art approaches in terms of classification results.

Multimodal multiobjective optimization problems (MMOPs) have recently received considerable attention since they emerge in many real-world applications (e.g., in multiobjective knapsack problems and flow shop scheduling). However, MMOPs constitute a very particular class of problem. Indeed, looking for an adequate Pareto front (PF) representation is insufficient. MMOPs contain multiple subsets within the Pareto optimal Set, each independently mapping to the same Pareto Front. So, traditional multiobjective evolutionary algorithms (MOEAs) are inappropriate for solving MMOPs. This has motivated the design of algorithms which are suitable for addressing MMOPs. This paper proposes modifying MMOEA/DC, which is an algorithm specifically designed for solving MMOPs, that adopts a dual clustering method in both decision and objective space. Particularly, we were interested in using clustering in decision space, which allows classifying solutions into multiple local clusters. Our proposed approach modifies the neighborhood definition in order to provide more robustness to the algorithm as well as to improve the clustering process in decision space and to reduce the influence of the control parameters. The efficiency of the proposed framework is validated by comparing its performance on test instances of two test suites (MMF and MMMPO) with respect to the original MMOEA/DC and other state-of-the-art algorithms designed for solving MMOPs.

Explicit and implicit averaging are two well-known strategies for noisy optimization. Both strategies can counteract the disruptive effect of noise; however, a critical question remains: which one is more efficient? This question has been raised in many studies, with conflicting preferences and, in some cases, findings. Nevertheless, theoretical findings on the noisy sphere problem with additive Gaussian noise supports the superiority of implicit averaging, which may have had a strong impact on the preference of implicit averaging in more recent evolutionary methods for noisy optimization. This study speculates that the analytically supported superiority of implicit averaging relies on specific features of the noisy sphere problem with additive noise, which cannot be generalized to other problems. It enumerates these features and designs controlled numerical experiments to investigate this potential reliance. Each experiment gradually suppresses one specific feature, and the progress rate is numerically calculated for different values of the sample size given a fixed evaluation budget. Our empirical results indicate that for a wide range of noise strength and evaluation budget per iteration, the more these specific features are suppressed, the more the optimal averaging strategy deviates from implicit toward explicit averaging, which confirms our speculations. Consequently, the optimal sample size, which is regarded as the tradeoff between implicit and explicit averaging, depends on the problem characteristics and should be learned during optimization for maximum efficiency.

Molecular docking plays a vital role in modern drug discovery, by supporting predictions of the binding modes and affinities of ligands at the binding site of target proteins. Several docking programs have been developed for both commercial and academic applications. Typically, a docking program’s performance depends on the sampling algorithm used to generate the ligand’s potential conformations and the scoring function applied to evaluate and rank these conformations. Evolutionary algorithms are widely used as sampling algorithms in docking programs. However, both the linkage problem and the dimensionality degenerate the search ability of evolutionary algorithms in the docking process. Therefore, a newly designed docking program named AutoDock Koto was developed in this study, which adopts a novel gradient boosting differential evolution algorithm to effectively address these issues. Experimental results show that compared with commonly used docking programs, AutoDock Koto yields dramatic improvements in docking performance based on an extensive dataset of 285 protein–ligand complexes. In addition, due to its strong docking ability, AutoDock Koto was used to identify potential drugs for COVID-19 based on a virtual screening of all approved drugs in our experiments. Sixteen drugs are found to possess low binding energy to the main target protease of SARS-CoV-2 and, thus, have the potential to treat COVID-19 as antiviral drugs. The source code of AutoDock Koto can be downloaded for free fromhttps://github.com/codezhouj/Molecular_Docking.

Discretization-based feature selection (DBFS) approaches have shown interesting results when using several metaheuristic algorithms, such as particle swarm optimization (PSO), genetic algorithm (GA), ant colony optimization (ACO), etc. However, these methods share the same shortcoming which consists in encoding the problem solution as a sequence of cut-points. From this cut-points vector, the decision of deleting or selecting any feature is induced. Indeed, the number of generated cut-points varies from one feature to another. Thus, the higher the number of cut-points, the higher the probability of selecting the considered feature; and vice versa. This fact leads to the deletion of possibly important features having a single or a low number of cut-points, such as the infection rate, the glycemia level, and the blood pressure. In order to solve the issue of the dependency relation between the feature selection (or removal) event and the number of its generated potential cut-points, we propose to model the DBFS task as a bilevel optimization problem and then solve it using an improved version of an existing co-evolutionary algorithm, named I-CEMBA. The latter ensures the variation of the number of features during the migration process in order to deal with the multimodality aspect. The resulting algorithm, termed bilevel discretization-based feature selection (Bi-DFS), performs selection at the upper level while discretization is done at the lower level. The experimental results on several high-dimensional datasets show that Bi-DFS outperforms relevant state-of-the-art methods in terms of classification accuracy, generalization ability, and feature selection bias.

Recently, a number of resource allocation strategies have been proposed for evolutionary algorithms to efficiently tackle multiobjective optimization problems (MOPs). However, these methods mainly allocate computational resources based on the convergence improvement under the decomposition-based framework, which may become ineffective with the increased number of optimization objectives. To address this problem, this article suggests an immune-inspired resource allocation strategy, which breaks through the decomposition-based framework and can better balance convergence and diversity for many-objective optimization. In our method, the diversity distances of solutions are defined by the Euclidean distances of their projected points on the unit hyperplane. Then, based on the diversity distances, resource allocation is realized by using an immune cloning operator to encourage exploring sparse regions of the search space. Moreover, to provide high-quality solutions in coordination with this immune cloning operator, a novel archive update mechanism is designed. When compared to most well-known resource allocation strategies, our method is advantageous for many-objective optimization. The experimental results also validate the superiority of our method over several state-of-the-art evolutionary algorithms for solving two sets of complicated MOPs having 5 to 15 objectives.

Cost-sensitive learning is one of the most adopted approaches to deal with data imbalance in classification. Unfortunately, the manual definition of misclassification costs is still a very complicated task, especially with the lack of domain knowledge. To deal with the issue of costs' uncertainty, some researchers proposed the use of intervals instead of scalar values. This way, each cost would be delimited by two bounds. Nevertheless, the definition of these bounds remains as a very complicated and challenging task. Recently, some researches proposed the use of genetic programming to simultaneously build classification trees and search for optimal costs' bounds. As for any classification tree there is a whole search space of costs' bounds, we propose in this paper a bi-level evolutionary approach for interval-based cost-sensitive classification tree induction where the trees are constructed at the upper level while misclassification costs intervals bounds are optimized at the lower level. This ensures not only a precise evaluation of each tree but also an effective approximation of optimal costs intervals bounds. The performance and merits of our proposal are shown through a detailed comparative experimental study on commonly used imbalanced benchmark data sets with respect to several existing works.

Most state-of-the-art Multiobjective Evolutionary Algorithms (moeas) promote the preservation of diversity of objective function space but neglect the diversity of decision variable space. The aim of this article is to show that explicitly managing the amount of diversity maintained in the decision variable space is useful to increase the quality of moeas when taking into account metrics of the objective space. Our novel Variable Space Diversity-based MOEA (vsd-moea) explicitly considers the diversity of both decision variable and objective function space. This information is used with the aim of properly adapting the balance between exploration and intensification during the optimization process. Particularly, at the initial stages, decisions made by the approach are more biased by the information on the diversity of the variable space, whereas it gradually grants more importance to the diversity of objective function space as the evolution progresses. The latter is achieved through a novel density estimator. The new method is compared with state-of-art moeas using several benchmarks with two and three objectives. This novel proposal yields much better results than state-of-the-art schemes when considering metrics applied on objective function space, exhibiting a more stable and robust behavior.

The most relevant property that a quality indicator (QI) is expected to have is Pareto compliance, which means that every time an approximation set strictly dominates another in a Pareto sense, the indicator must reflect this. The hypervolume indicator and its variants are the only unary QIs known to be Pareto-compliant but there are many commonly used weakly Pareto-compliant indicators such as R2, IGD+, and ε+. Currently, an open research area is related to finding new Pareto-compliant indicators whose preferences are different from those of the hypervolume indicator. In this article, we propose a theoretical basis to combine existing weakly Pareto-compliant indicators with at least one being Pareto-compliant, such that the resulting combined indicator is Pareto-compliant as well. Most importantly, we show that the combination of Pareto-compliant QIs with weakly Pareto-compliant indicators leads to indicators that inherit properties of the weakly compliant indicators in terms of optimal point distributions. The consequences of these new combined indicators are threefold: (1) to increase the variety of available Pareto-compliant QIs by correcting weakly Pareto-compliant indicators, (2) to introduce a general framework for the combination of QIs, and (3) to generate new selection mechanisms for multiobjective evolutionary algorithms where it is possible to achieve/adjust desired distributions on the Pareto front.

Performance of multi/many-objective evolutionary algorithms (MOEAs) based on decomposition is highly impacted by the Pareto front (PF) shapes of multi/many-objective optimization problems (MOPs), as their adopted weight vectors may not properly fit the PF shapes. To avoid this mismatch, some MOEAs treat solutions as weight vectors to guide the evolutionary search, which can adapt to the target MOP's PF automatically. However, their performance is still affected by the similarity metric used to select weight vectors. To address this issue, this article proposes a fuzzy decomposition-based MOEA. First, a fuzzy prediction is designed to estimate the population's shape, which helps to exactly reflect the similarities of solutions. Then, N least similar solutions are extracted as weight vectors to obtain N constrained fuzzy subproblems ( N is the population size), and accordingly, a shared weight vector is calculated for all subproblems to provide a stable search direction. Finally, the corner solution for each of m least similar subproblems ( m is the objective number) is preserved to maintain diversity, while one solution having the best aggregated value on the shared weight vector is selected for each of the remaining subproblems to speed up convergence. When compared to several competitive MOEAs in solving a variety of test MOPs, the proposed algorithm shows some advantages at fitting their different PF shapes.

Generally, decomposition-based evolutionary algorithms in many-objective optimization (MaOEA/Ds) have widely used reference vectors (RVs) to provide search directions and maintain diversity. However, their performance is highly affected by the matching degree on the shapes of the RVs and the Pareto front (PF). To address this problem, this article proposes a self-guided RV (SRV) strategy for MaOEA/Ds, aiming to extract RVs from the population using a modified k -means clustering method. To give a promising clustering result, an angle-based density measurement strategy is used to initialize the centroids, which are then adjusted to obtain the final clusters, aiming to properly reflect the population's distribution. Afterward, these centroids are extracted to obtain adaptive RVs for self-guiding the search process. To verify the effectiveness of this SRV strategy, it is embedded into three well-known MaOEA/Ds that originally use the fixed RVs. Moreover, a new strategy of embedding SRV into MaOEA/Ds is discussed when the RVs are adjusted at each generation. The simulation results validate the superiority of our SRV strategy, when tackling numerous many-objective optimization problems with regular and irregular PFs.

Existing solution approaches for handling disruptions in project scheduling use either proactive or reactive methods. However, both techniques suffer from some drawbacks that affect the performance of the optimization process in obtaining good quality schedules. Therefore, in this article, we develop an auto-configured multioperator evolutionary approach, with a novel pro-reactive scheme for handling disruptions in multimode resource-constrained project scheduling problems (MM-RCPSPs). In this article, our primary objective is to minimize the makespan of a project. However, we also have secondary objectives, such as maximizing the free resources (FRs) and minimizing the deviation of activity finishing time. As the existence of FR may lead to a suboptimal solution, we propose a new operator for the evolutionary approach and two new heuristics to enhance the algorithm's performance. The proposed methodology is tested and analyzed by solving a set of benchmark problems, with its results showing its superiority with respect to state-of-the-art algorithms in terms of the quality of the solutions obtained.

The covariance matrix self-adaptation evolution strategy with repelling subpopulations (RS-CMSA-ES) is one of the most successful multimodal optimization (MMO) methods currently available. However, some of its components may become inefficient in certain situations. This study introduces the second variant of this method, called RS-CMSA-ESII. It improves the adaptation schemes for the normalized taboo distances of the archived solutions and the covariance matrix of the subpopulation, the termination criteria for the subpopulations, and the way in which the infeasible solutions are treated. It also improves the time complexity of RS-CMSA-ES by updating the initialization procedure of a subpopulation and developing a more accurate metric for determining critical taboo regions. The effects of these modifications are illustrated by designing controlled numerical simulations. RS-CMSA-ESII is then compared with the most successful and recent niching methods for MMO on a widely adopted test suite. The results obtained reveal the superiority of RS-CMSA-ESII over these methods, including the winners of the competition on niching methods for MMO in previous years. Besides, this study extends RS-CMSA-ESII to dynamic MMO and compares it with a few recently proposed methods on the modified moving peak benchmark functions.

Surrogate-assisted evolutionary algorithms (SAEAs) have become very popular for tackling computationally expensive multiobjective optimization problems (EMOPs), as the surrogate models in SAEAs can approximate EMOPs well, thereby reducing the time cost of the optimization process. However, with the increased number of decision variables in EMOPs, the prediction accuracy of surrogate models will deteriorate, which inevitably worsens the performance of SAEAs. To deal with this issue, this article suggests an ensemble surrogate-based framework for tackling EMOPs. In this framework, a global surrogate model is trained under the entire search space to explore the global area, while a number of surrogate submodels are trained under different search subspaces to exploit the subarea, so as to enhance the prediction accuracy and reliability. Moreover, a new infill sampling criterion is designed based on a set of reference vectors to select promising samples for training the models. To validate the generality and effectiveness of our framework, three state-of-the-art evolutionary algorithms [nondominated sorting genetic algorithm III (NSGA-III), multiobjective evolutionary algorithm based on decomposition with differential evolution (MOEA/D-DE) and reference vector-guided evolutionary algorithm (RVEA)] are embedded, which significantly improve their performance for solving most of the test EMOPs adopted in this article. When compared to some competitive SAEAs for solving EMOPs with up to 30 decision variables, the experimental results also validate the advantages of our approach in most cases.

Many real systems are represented in form of multiplex networks composed of a set of nodes, multiple layers of links, and coupling node relationships across all layers. These systems are very vulnerable to damages during both attacks and recoveries due to potential node cascading failures (NCFs). Although some progress has recently been made in studying network robustness and resilience, the comprehensive impacts of coupling node relationships and community structures on NCFs remain unclear. Accordingly, in this article, we study the robustness and resilience of multiplex networks in the presence of NCFs caused by coupling node relationships and community structures. We first model the failure processes of multiplex networks during both attacks and recoveries as node-community cascading failures (called NCCFs), and then theoretically demonstrate the fragility of multiplex networks to random node damages under NCCFs. Subsequently, to improve network robustness and resilience, we adopt a node protection strategy and propose a cost-aware constrained optimization problem. Finally, we devise a degree-based simulated annealing algorithm for solving this optimization problem. Extensive experiments on both simulated and real multiplex networks show that NCCFs make networks more vulnerable to unpredictable damage than classical NCFs. The results also show the superiority of the proposed algorithm over the state-of-the-art algorithms in improving network robustness and resilience.

There has been a considerable amount of research on the development of metaheuristic methods for resource-constrained project scheduling problems. Early methods followed the building blocks and even the formulation of well-understood metaheuristic methods as well as simple but effective heuristics such as forward-backward improvement. In contrast, more recent methods employ less familiar, more complex (hybrid) metaheuristics and non-standard components and formulations. Although the former may provide better results on standard test problems, it is not easy to understand how each component has contributed to improving the results and why a deviation from well-established formulations, components and methods was necessary. This research advances our knowledge about the impact of different strategies and components of customized genetic algorithms (some of which have been proposed in this study) on the optimization results. This task is performed by developing a comprehensive genetic algorithm with several familiar and potentially effective components. A modular analysis is then performed in which one component is suppressed at a time, and the resultant performance decline is analyzed. With hindsight from the modular analysis, a simple method is suggested and the importance of each component is clarified. Thus, no further simplification can be performed without compromising efficiency. Our preliminary results reveal that this customized genetic algorithm outperforms many existing methods and can compete with the most successful ones, which, in many cases, are much more complex than our approach.

Generative Adversarial Network (GAN) is a family of machine learning algorithms designed to train neural networks able to imitate real data distributions. Unfortunately, GAN suffers from problems such as gradient vanishing and mode collapse. In Multi-Objective Evolutionary Generative Adversarial Network (MO-EGAN) these problems were addressed using an evolutionary technique combined with Multi-Objective selection, obtaining better results on synthetic datasets at the expense of larger computation times. In this works, we present the Smart MultiObjective Evolutionary Generative Adversarial Network (SMO-EGAN) algorithm, which reduces the computational cost of MO-EGAN and achieves better results on real data distributions.

The hypervolume remains a popular performance indicator in evolutionary multi-objective, mainly because of its nice mathematical properties (i.e., it's the only performance indicator known to be Pareto-compliant). However, its high computational cost (which grows polynomially on the population size but exponentially on the number of objectives) has severely limited its use in many-objective optimization. This has motivated a variety of proposals that attempt to overcome this limitation. One of the most popular proposals currently available is the so-called Hypervolume by Slicing Objectives (HSO) algorithm. Here, we show that the worst-case time complexity of the HSO algorithm, as obtained by its authors, is incorrect. Then, we provide an efficient implementation of the HSO algorithm, which guarantees that unique slices are generated to compute the hypervolume.

Mating restrictions have been used to improve the performance of Multi-Objective Evolutionary Algorithms (MOEAs) by altering the way in which parents are selected in the recombination step. Originally proposed for single-objective optimization, mating restrictions have been implemented in different MOEAs obtaining mixed results. However, the role of mating restrictions in diversity management/maintenance and in the proper balance between exploration and exploitation within MOEAs has not been studied in sufficient detail in spite of its evident importance. In this paper, we present an empirical study on the impact of three new mating restrictions based on the s-energy performance indicator. When obtaining each individual's contribution to the total s-energy, we implicitly obtain vicinity information, since a high contribution means that an individual is relatively close to at least some other individual, i.e., it is in a crowded region. Conversely, an individual with a low contribution is in a non-crowded region. Using this information we explore different strategies aiming to improve the diversity of the population during its execution, as well as exploiting the least crowded regions of the objective space. One of the main advantages of our proposal are both its simplicity and its ability to scale up (in objective function space). We evaluate the impact of our proposals by implementing them in NSGA-III and comparing the obtained results with respect to those of the original algorithm. Our experimental results show that the use of mating restrictions does provide improvements in most of the test instances adopted for some of our proposed strategies.

Ensembles have been used in the evolutionary computation literature to evolve several populations in an independent manner, using different search approaches. Moreover, each population’s parents compete with their offspring and the other population’s offspring to improve diversity. It has been shown that ensemble algorithms improve the performance of the techniques embedded within them, when considered independently. Furthermore, scalarizing functions have been successfully used in decomposition-based and some indicator-based Multi-objective Evolutionary Algorithms (MOEAs). However, it has been shown that the performance of scalarizing function tends to be tied to the geometrical shape of the Pareto front. In this work, we propose a new ensemble algorithm that adopts different scalarizing functions and weight vectors using Hungarian Differential Evolution as the baseline multi-objective optimizer. Our experimental study shows that our proposed approach outperforms the original HDE, and it is competitive with respect to modern MOEAs.

Traditional reproduction operators in many-objective evolutionary algorithms (MaOEAs) seem to not be so effective to tackle many-objective optimization problems (MaOPs). This is mainly because the population size cannot be set to an arbitrarily large value if the computational efficiency is of concern. In such a case, the distance between the parents becomes remarkably large and, consequently, it is not easy to reproduce a superior offspring in high-dimensional objective space. To alleviate this problem, an elite gene-guided (EGG) reproduction operator is proposed to tackle MaOPs in this article. In this operator, an elite gene pool is built by collecting the knee points from the current population. Then, the offspring is produced by exchanging the genes with this elite gene pool under an exchange rate, aiming to reserve more promising genes into the next generation. In order to provide new genes for the population, other genes will be disturbed under a disturbance rate. The settings and functional analysis of the exchange rate and disturbance rate are studied using several experiments. The proposed EGG operator is easy to implement and can be embedded to any MaOEA. As examples, we show the embedding of the proposed EGG operator into four competitive MaOEAs, that is, MOEA/D, NSGA-III, θ -DEA, and SPEA2-SDE provide some advantages over simulated binary crossover, differential evolution, and an evolutionary path-based reproduction operator on solving a number of benchmark problems with 3 to 15 objectives.

This study develops an adaptive multilevel prediction (AMLP) method to detect and track multiple global optima over time. First, it formulates a multilevel prediction approach in which a higher level prediction improves the accuracy of the lower level prediction to reduce the prediction error, enabling it to capture more complex patterns in the changes. However, a higher level prediction is more sensitive to input errors and the randomness in the pattern of the change. To overcome this challenge, this study employs an adaptive mechanism which can determine the near-optimal prediction level at each time step. At the same time, AMLP calculates the strength of the diversity introduced after a change based on the estimated prediction error. A successful static multimodal optimizer is augmented with AMLP, for which AMLP determines the location and the mutation strength of the initialized subpopulations. An existing dynamic benchmark generator is improved so that it can generate dynamic test problems with more complex patterns in their changes. In particular, this dynamic benchmark generator allows for controlling the randomness of the pattern in the change to simulate dynamic problems with different degrees of predictability. A few controlled experiments are first performed to provide insight into different components of AMLP. Then, AMLP is compared with some of the most successful prediction methods when they are incorporated into the developed dynamic multimodal optimization method. Eleven dynamic cases with different change severity, change frequency, predictability, problem dimensionality, and the number of global minima are considered. The numerical results show the superiority of AMLP over other prediction methods.

For almost 20 years, quality indicators (QIs) have promoted the design of new selection mechanisms of multiobjective evolutionary algorithms (MOEAs). Each indicator-based MOEA (IB-MOEA) has specific search preferences related to its baseline QI, producing Pareto front approximations with different properties. In consequence, an IB-MOEA based on a single QI has a limited scope of multiobjective optimization problems (MOPs) in which it is expected to have a good performance. This issue is emphasized when the associated Pareto front geometries are highly irregular. In order to overcome these issues, we propose here an island-based multiindicator algorithm (IMIA) that takes advantage of the search biases of multiple IB-MOEAs through a cooperative scheme. Our experimental results show that the cooperation of multiple IB-MOEAs allows IMIA to perform more robustly (considering several QIs) than the panmictic versions of its baseline IB-MOEAs as well as several state-of-the-art MOEAs. Additionally, IMIA shows a Pareto-front-shape invariance property, which makes it a remarkable optimizer when tackling MOPs with complex Pareto front geometries.

This article suggests a multimodal multiobjective evolutionary algorithm with dual clustering in decision and objective spaces. One clustering is run in decision space to gather nearby solutions, which will classify solutions into multiple local clusters. Nondominated solutions within each local cluster are first selected to maintain local Pareto sets, and the remaining ones with good convergence in objective space are also selected, which will form a temporary population with more than${N}$solutions (${N}$is the population size). After that, a second clustering is run in objective space for this temporary population to get${N}$final clusters with good diversity in objective space. Finally, a pruning process is repeatedly run on the above clusters until each cluster has only one solution, which removes the most crowded solution in decision space from the most crowded cluster in objective space each time. This way, the clustering in decision space can distinguish all Pareto sets and avoid the loss of local Pareto sets, while that in objective space can maintain diversity in objective space. When solving all the benchmark problems from the competition of multimodal multiobjective optimization in the IEEE Congress on Evolutionary Computation 2019, the experiments validate our advantages to maintain diversity in both objective and decision spaces.

Multi-objective evolutionary algorithms (MOEAs) of the state of the art are created with the only purpose of dealing with the number of objective functions in a multi-objective optimization problem (MOP) and treat the decision variables of a MOP as a whole. However, when dealing with MOPs with a large number of decision variables (more than 100) their efficacy decreases as the number of decision variables of the MOP increases. On the other hand, problem decomposition, in terms of decision variables, has been found to be extremely efficient and effective for solving large scale optimization problems. Nevertheless, most of the currently available approaches for large scale optimization rely on models based on cooperative coevolution or linkage learning methods that use multiple subpopulations or preliminary analysis, respectively, which is computationally expensive (in terms of function evaluations) when used within MOEAs. In this work, we study the effect of what we call operational decomposition, which is a novel framework based on coevolutionary concepts to apply MOEAs's crossover operator without adding any extra cost. We investigate the improvements that NSGA-III can achieve when combined with our proposed coevolutionary operators. This new scheme is capable of improving efficiency of a MOEA when dealing with large scale MOPs having from 200 up to 1200 decision variables.

One of the challenges that appear in solving constrained optimization problems is to quickly locate the search areas of interest. Although the initial solutions of any optimization algorithm have a significant effect on its performance, none of the existing initialization methods can provide direct information about the objective function and constraints of the problem to be solved. In this paper, a technique for generating initial solutions is proposed, which provides useful information about the behavior of both the objective function and the constraints. Based on such information, an automatic mechanism for selecting individuals, from the search areas of interest, is introduced. The proposed method is adopted with different evolutionary algorithms and tested on the CEC2006 and the CEC2010 test problems. The results obtained show the benefits of the proposed method in enhancing the performance, and reducing the average computational time, of several algorithms with respect to their versions adopting other initialization techniques.

Currently, reference sets, which are a collection of feasible or infeasible points in objective space, are the backbone of several multi-objective evolutionary algorithms (MOEAs) and quality indicators (QIs). For both MOEAs and QIs, an important question is how to construct the reference set regardless of the dimensionality of the objective space, preserving well-diversified solutions. The Simplex-Lattice-Design method (SLD) that constructs a set of convex weights in a simplex, has been usually used to define reference sets. However, it is not a good option since Pareto fronts with irregular geometries cannot be completely intersected by the weight vectors. In this paper, we propose a tool based on the Riesz s-energy to generate reference sets exhibiting good diversity properties. Our experimental results support the Riesz s-energy-based reference sets as a better option due to their invariance to the Pareto front shape and the objective space dimensionality.

The robust controllability (RC) of a complex system tries to select a set of dominating entities for the functional control of this entire system without uncertain disturbances, and the research on RC will help to understand the system's underlying functions. In this article, we introduce the control cost in signed networks and present a cost-aware robust control (CRC) problem in this scenario. The aim of CRC is to minimize the cost to control a set of dominating nodes and transform a set of unbalanced links into balanced ones, such that the signed network can be robustly controlled without uncertain unbalanced factors (like nodes and links). To solve this problem, we first model CRC as a constrained combination optimization problem, and then present a memetic algorithm with some problem-specific knowledge (like the neighbors of nodes, the constraints of CRC, and the fast computation of the cost under each optimization) to solve this problem on signed networks. The extensive experiments on both real social and biological networks assess that our algorithm outperforms several state-of-the-art RC algorithms.

Decomposition-based evolutionary algorithms using predefined reference points have shown good performance in many-objective optimization. Unfortunately, almost all experimental studies have focused on problems having regular Pareto fronts (PFs). Recently, it has been shown that the performance of such algorithms is deteriorated when facing irregular PFs, such as degenerate, discontinuous, inverted, strongly convex, and/or strongly concave fronts. The main issue is that the predefined reference points may not all intersect with the PF. Therefore, many researchers have proposed to update the reference points with the aim of adapting them to the discovered Pareto shape. Unfortunately, the adaptive update does not really solve the issue for two main reasons. On the one hand, there is a considerable difficulty to set the time and the frequency of updates. On the other hand, it is not easy to define how to update the search directions for an unknown PF shape. This article proposes to approximate irregular PFs using a set of predefined normal-boundary intersection (NBI) directions. The main motivation behind this article is that when using a set of well-distributed NBI directions, all these directions intersect with the PF regardless of its shape, except for the case of discontinuous and/or degenerate fronts. To handle the latter cases, a simple interaction mechanism between the decision maker (DM) and the algorithm is used. In fact, the DM is asked if the number of NBI directions needs to be increased in some stages of the evolutionary process. If so, the resolution of the NBI directions that intersect the PF is increased to properly cover discontinuous and/or degenerate PFs. Our experimental results on benchmark problems with regular and irregular PFs, having up to fifteen objectives, show the merits of our algorithm when compared to eight of the most representative state-of-the-art algorithms.

Black-box topology optimization (BBTO) uses evolutionary algorithms and other soft computing techniques to generate near-optimal topologies of mechanical structures. Although evolutionary algorithms are widely used to compensate the limited applicability of conventional gradient optimization techniques, methods based on BBTO have been criticized due to numerous drawbacks. In this article, we discuss topology optimization as a black-box optimization problem. We review the main BBTO methods, discuss their challenges and present approaches to relax them. Dealing with those challenges effectively can lead to wider applicability of topology optimization, as well as the ability to tackle industrial, highly constrained, nonlinear, many-objective, and multimodal problems. Consequently, future research in this area may open the door for innovating new applications in science and engineering that may go beyond solving classical optimization problems of mechanical structures. Furthermore, algorithms designed for BBTO can be added to existing software toolboxes and packages of topology optimization.

In recent years, several indicator-based multi-objective evolutionary algorithms (IB-MOEAs) have been proposed. Each IB-MOEA presents different search preferences depending on the quality indicator (QI) that it uses in its selection mechanism. However, due to these search biases, IB-MOEAs behave differently on each multi-objective optimization problem, producing Pareto front approximations whose characteristics are related to the QI on which they are based. In this paper, we propose a novel algorithm based on the island model that aims to take advantage of the cooperation of individual IB-MOEAs based on the indicators hypervolume, R2, IGD+,+, and Δpwith the aim of improving both convergence and distribution of the Pareto fronts produced. Our experimental results, taking into account seven quality indicators, empirically show that the cooperation of several IB-MOEAs is better than using panmictic versions of them. Additionally, we also show that the performance of our proposal does not depend on the Pareto front shape of the problem being solved.

In this paper, a simple and effective termination condition for both single- and multi-objective evolutionary algorithms has been proposed. The termination condition is based on simply observing objective values of solution candidates during generations. Effectiveness of the termination condition is self-evident with single-objective problems but unclear with multi-objective problems. Therefore, experiments with some well known bi- and tri-objective test problems have been performed. The proposed termination condition is implemented in Generalized Differential Evolution (GDE) that is a general purpose optimization algorithm for both single- and multi-objective optimization with or without constraints. Our preliminary results indicate that the proposed termination condition is a suitable termination condition also with multi-objective problems. With the termination condition and a control parameter adaptation technique previously introduced, GDE has become a fully automated optimization algorithm that can be used by any optimization practitioner.

In evolutionary multi-objective optimization, the results generated by an evolutionary algorithm usually contain an approximation, as good as possible, of the entire Pareto-optimal front. However, sometimes the number of Pareto-optimal solutions may be so large that the decision maker (DM) is incapable of manipulating or understanding them. Methods for considering only the Pareto-optimal solutions that the DM prefers indeed constitute a hot research topic in the evolutionary computation field. In this paper, we introduce a new dominance relation called $\hat g$-dominance, which is an improved version of the g-dominance relation and can be easily implemented in traditional multi-objective evolutionary algorithms. In this work, the proposed $\hat g$-dominance is implemented in NSGA-II. Our experimental results show the effectiveness of $\hat g$-NSGA-II with respect to the original g-NSGA-II.

One of the bottlenecks in steady-state multiobjective evolutionary algorithms (MOEAs) is non-dominated sorting because it is performed every time whenever a new offspring is generated. The recent literature shows that there is no requirement to perform the complete non-dominated sorting procedure because the entire structure of non-domination level (NDL) does not change. Some approaches have been recently proposed based on this idea. In this paper, we update our previous work where an offspring is inserted into the set of fronts, to further reduce the number of dominance comparisons. Additionally, we also explore parallelism in the updated approach in two different manners considering the PRAM CREW model. Finally, the time and space complexities of two parallel versions is theoretically analyzed.

In the current paper we focus on parallelization of non-dominated sorting which is an essential step in Pareto-based multi-objective evolutionary algorithms. The parallel approaches can help to reduce the overall execution time of multi-objective evolutionary algorithms. Although there have been some proposals to parallelize non-dominated sorting algorithms, most of them have focused on the fast non-dominated sort algorithm proposed by Deb et al. This paper explores the scope of parallelism in a recently proposed approach known as Best Order Sort, which was proposed by Roy et al. We focus on two different ways of achieving parallelism in Best Order Sort. The time and space complexity of these two parallel schemes is also analyzed theoretically considering the PRAM CREW model.

Over the last few decades, project scheduling problems have been solved under a set of resource constraints, which are assumed fixed throughout the project horizon. However, in real-life applications, resources may change over time due to maintenance or because the resources are needed for another project. Therefore, this research introduces a hybrid evolutionary framework, based on two multi-operator evolutionary algorithms, and a heuristic technique, for a multi-mode project scheduling under irregular resources changes. The framework simultaneously considers both algorithms and self-adaptively emphasizes the one which performs comparatively better. The heuristic considers two variants of handling techniques for irregular resources. One is based on inserting buffer activities to characterize resources unavailable, and another is based on a modified serial generation scheme, that determines the best modes of the activities at each time period based on irregular resources. The framework is tested by solving a set of test problems, with the renewable resources considered irregular over the project horizon. The results demonstrate that the multi-method algorithm has some advantages for scheduling a project, under both regular and irregular resources.

In recent years, quality indicators (QIs) have been employed to design selection mechanisms for multi-objective evolutionary algorithms (MOEAs). These indicator-based MOEAs (IB-MOEAs) generate Pareto front approximations that present convergence and diversity characteristics strongly related to the QI that guides the selection mechanism. However, on complex multi-objective optimization problems, the performance of IB-MOEAs is far from being completely understood. In this paper, we empirically analyze the convergence and diversity properties of five steady-state IB-MOEAs based on the hypervolume, R2, IGD+, ∈+, and Δp. Regarding convergence, we analyze their speed of convergence and the final closeness to the true Pareto front. The IB-MOEAs adopted in our study were tested on problems having different Pareto front shapes, and were taken from six test suites. Our experimental results show general and particular strengths and weaknesses of the adopted IB-MOEAs. We believe that these results are the first step towards a deeper understanding of the behavior of IB-MOEAs.

The assessment of strengths and weaknesses of a solver is often limited by the diversity of the cases where it is tested upon. As such, it is paramount to have a versatile tool which finds the problem instances where such a solver excels/fails. In this manuscript, we propose to use an evolutionary algorithm for creating this tool. To validate our approach, we conducted several tests on four heuristics for the knapsack problem. Although, the process can be extended to other domains with relatively few changes. The tests cover different sets of instances, both favoring the performance of one heuristic while hindering that of the remaining ones, and vice versa. To further test our evolutionary-based model, we also apply it on a recent approach that combines the strengths of different heuristics to improve its performance (usually referred to as a hyper-heuristic). We show that it is possible to tailor instances in which even this more complex model excels/fails. Throughout our approach, a researcher can test a solver under different kinds of scenarios, delving deeper into the conditions that make it perform well/poorly. Therefore, we recommend using the proposed approach as a means to grasp better insights about strengths and weaknesses of different solvers.

During the last two decades, many multioperator- and multimethod-based evolutionary algorithms for solving optimization problems have been proposed. Although, in general terms, they outperform single-operator-based traditional ones, they do not perform consistently for all the problems tested in the literature. The designs of such algorithms usually follow a trial and error approach that can be improved by using a rule-based approach. In this paper, we propose a new way for two algorithms to cooperate as an effective team, in which a heuristic is applied using fuzzy rules of two complementary characteristics, the quality of solutions and diversity in the population. In this process, two subpopulations are used, one for each algorithm, with greater emphasis placed on the better-performing one. Inferior algorithms learn from trusted ones and a fine-tuning procedure is applied in the later stages of the evolutionary process. The proposed algorithm was analyzed on the CEC2014 unconstrained problems and then tested on other three sets (CEC2013, CEC2005, and 12 classical problems), with its results showing a high success rate and that it outperformed both single-operator-based and different state-of-the-art algorithms.

Complex systems are often characterized by complex networks with links and entities. However, in many complex systems such as protein-protein interaction networks, recommender systems, and online communities, their links are hard to reveal directly, but they can be inaccurately observed by multiple data collection platforms or by a data collection platform at different times. Then, the links of the systems are inferred by the integration of the collected observations. As those data collection platforms are usually distributed over a large area and in different fields, their observations are unreliable and sensitive to the potential structures of the systems. In this paper, we consider the link inference problem in network data with community structures, in which the reliability of data collection platforms is unknown a priori and the link errors and reliability of platforms' observations are heterogeneous to the underlying community structures of the systems. We propose an expectation maximization algorithm for link inference in a network system with community structures (EMLIC). The EMLIC algorithm is also used to infer the link errors and reliability of platforms' observations in different communities. Experimental results on both synthetic data and eight real-world network data demonstrate that our algorithm is able to achieve lower link errors than the existing reliable link inference algorithms when the network data have community structures.

This paper suggests a novel clustering-based evolutionary algorithm for many-objective optimization problems. Its main idea is to classify the population into a number of clusters, which is expected to solve the difficulty of balancing convergence and diversity in high-dimensional objective space. The individuals showing high similarities on the vector angles are gathered into the same cluster, such that the population’s distribution can be well portrayed by the clusters. To efficiently find these clusters, partitional clustering is first used to classify the union population into${m}$main clusters based on the${m}$axis vectors (${m}$is the number of objectives), and then hierarchical clustering is further run on these${m}$main clusters to get${N}$final clusters (${N}$is the population size and${N>m}$). At last, in environmental selection, one individual from each of${N}$clusters closest to the axis vectors is selected to maintain diversity, while one individual from each of the other clusters is preferred by a simple convergence indicator to ensure convergence. When tackling some well-known test problems with 5–15 objectives, extensive experiments validate the superiority of our algorithm over six competitive many-objective EAs, especially on problems with incomplete and irregular Pareto-optimal fronts.

In multiobjective optimization, a scalable test problem is one that can be formulated for an arbitrary number of objectives. Scalable test problems evaluate the conceptual foundations of the so-called many-objective evolutionary algorithms. As an important class of problems, scalable test problems should contemplate a wide variety of features allowing us to evaluate and judge specific components of many-objective evolutionary algorithms. This, in fact, should promote the development of new strategies and/or methods in the design of many-objective optimization approaches. For this reason, the study of features and difficulties of this class of problems, plays a salient role in the development of many-objective approaches. As a result, a number of multiobjective scalable test problems have been proposed in recent years. In this paper, we present a review of features and limitations of existing multiobjective test problems formulated in continuous and unconstrained search spaces. We examine some features observed in some test problems which have not been properly discussed before. Additionally, we summarize a list of features and recommendations that should be considered in the design of scalable multiobjective test instances. Then, we preset a review of the state-of-the-art scalable test suites, including their features and limitations according to the recommended guidelines discussed herein. Finally, some possible paths for future research in this area are briefly discussed.

This paper proposes an effective ensemble framework (EF) for tackling multiobjective optimization problems, by combining the advantages of various evolutionary operators and selection criteria that are run on multiple populations. A simple ensemble algorithm is realized as a prototype to demonstrate our proposed framework. Two mechanisms, namely competition and cooperation, are employed to drive the running of the ensembles. Competition is designed by adaptively running different evolutionary operators on multiple populations. The operator that better fits the problem’s characteristics will receive more computational resources, being rewarded by a decomposition-based credit assignment strategy. Cooperation is achieved by a cooperative selection of the offspring generated by different populations. In this way, the promising offspring from one population have chances to migrate into the other populations to enhance their convergence or diversity. Moreover, the population update information is further exploited to build an evolutionary potentiality model, which is used to guide the evolutionary process. Our experimental results show the superior performance of our proposed ensemble algorithms in solving most cases of a set of 31 test problems, which corroborates the advantages of our EF.

In recent years, several non-dominated sorting approaches have been proposed. Non-dominated sorting is an essential part of Pareto dominance-based multi-objective evolutionary algorithms (MOEAs) and therefore the relevance of being able to perform such process as efficiently as possible is important. As the use of parallelism has become increasingly popular within MOEAs, there is an evident need for parallel implementations of non-dominated sorting algorithms. In this paper, we have focused on an efficient non-dominated sorting (ENS) approach and explored its parallelization. The time complexity of the parallel version of ENS is theoretically analyzed in four different scenarios.

In recent years, the use of decomposition-based multi-objective evolutionary algorithms has been very successful in solving both multi- and many-objective optimization problems. In these algorithms, the adopted Scalarizing Functions (SFs) play a crucial role in their performance. Methods such as the Modified Weighted Chebyshev (MCHE), Penalty Boundary Intersection (PBI) and Augmented Achievement Scalarizing Function (AASF) have been found to be very effective for achieving both convergence to the true Pareto front and a uniform distribution of solutions along it. However, the choice of an appropriate model parameter is required for these SFs. Some studies have analyzed the impact of these parameter values on the performance of the best-known decomposition multi-objective evolutionary algorithm (MOEA/D). In this paper, we propose a strategy based on collaborative populations combining different SFs and model parameter values via an adaptive operator selection based on the multi-armed bandit technique. Our preliminary results give rise to some interesting observations regarding the way in which different SFs are combined and adapted during the evolutionary process of MOEA/D.

In recent years, Indicator-based Multi-Objective Evolutionary Algorithms (IB-MOEAs) have become a relatively popular alternative for solving multi-objective optimization problems. IB-MOEAs are normally based on the use of a single performance indicator. However, the effect of the combination of multiple performance indicators for selecting solutions is a topic that has rarely been explored. In this paper, we propose a hyper-heuristic which combines the strengths and compensates for the weaknesses of four density estimators based on R2, IGD+, ϵ+ and Δp. The selection of the indicator to be used at a particular moment during the search is done using online learning and a Markov chain. Additionally, we propose a novel framework that aims to reduce the computational cost involved in the calculation of the indicator contributions. Our experimental results indicate that our proposed approach can outperform state-of-the-art MOEAs based on decomposition (MOEA/D) reference points (NSGA-III) and the R2 indicator (R2-EMOA) for problems with both few and many objectives.

In recent years, the design of new selection mechanisms has become a popular trend in the development of Multi-Objective Evolutionary Algorithms (MOEAs). This trend has been motivated by the aim of maintaining a good balance between convergence and diversity of the solutions. Reference-based selection is, with no doubt, one of the most promising schemes in this area. However, reference-based MOEAs are known to have difficulties for solving multi-objective problems with complicated Pareto fronts, mainly because they rely on the consistency between the Pareto front shape and the distribution of the reference weight vectors. In this paper, we propose a reference-based MOEA, which uses the Inverted Generational Distance plus (IGD+) indicator. The proposed approach adopts a novel method for approximating the reference set, based on an hypercube-based method. Our results indicate that our proposed approach is able to obtain solutions of a similar quality to those obtained by RVEA, MOEA/DD, NSGA-III and MOMBI-II in several test problems traditionally adopted in the specialized literature, and is able to outperform them in problems with complicated Pareto fronts.

In recent years, evolutionary algorithms have been found to be effective and efficient techniques to train support vector machines (SVMs) for binary classification problems while multiclass problems have been neglected. This paper proposes CMOE-SVM: Cooperative Multi-Objective Evolutionary SVMs for multiclass problems. CMOE-SVM enables SVMs to handle multiclass problems via co-evolutionary optimization, by breaking down the original M-class problem into M simpler ones, which are optimized simultaneously in a cooperative manner. Furthermore, CMOE-SVM can explicitly maximize the margin and reduce the training error (the two components of the SVM optimization), by means of multi-objective optimization. Through a comprehensive experimental evaluation using a suite of benchmark datasets, we validate the performance of CMOE-SVM. The experimental results, supported by statistical tests, give evidence of the effectiveness of the proposed approach for solving multiclass classification problems.

The multiobjective evolutionary algorithm (MOEA) based on decomposition transforms a multiobjective optimization problem into a set of aggregated subproblems and then optimizes them collaboratively. Since these subproblems usually have different degrees of difficulty, resource allocation (RA) strategies have been reported to enhance performance, attempting to dynamically assign proper amounts of computational resources for the solution of each of these subproblems. However, existing schemes for decomposition-based MOEAs fully rely on the relative improvement of the aggregated functions to do this. This paper proposes a diversity-enhanced RA strategy for this kind of MOEA, depending on both relative improvement on aggregated function value and solution density around each subproblem to assign computational resources. Thus, one subproblem surrounded with fewer solutions in its neighboring area and more relative improvement on the aggregated function value will be allocated a higher probability for evolution. Our experimental results show the advantages of our proposed strategy over two popular RA strategies available for decomposition-based MOEAs, on tackling a set of complicated benchmark problems.

In the last 20 years, evolutionary algorithms (EAs) have shown to be an effective method to solve multiobjective optimization problems (MOPs). Due to their population-based nature, multiobjective EAs (MOEAs) are able to generate a set of tradeoff solutions (called nondominated solutions) in a single algorithmic execution instead of having to perform a series of independent executions, as normally done with mathematical programming techniques. Additionally, MOEAs can be successfully applied to problems with difficult features such as multifrontality, discontinuity and disjoint feasible regions, among others. On the other hand, coevolutionary algorithms (CAs) are extensions of traditional EAs which have become subject of numerous studies in the last few years, particularly for dealing with large-scale global optimization problems. CAs have also been applied to the solution of MOPs, motivating the development of new algorithmic and analytical formulations that have advanced the state-of-the-art in CAs research, while simultaneously opening a new research path within MOEAs. This paper presents a critical review of the most representative coevolutionary MOEAs (CMOEAs) that have been reported in the specialized literature. This survey includes a taxonomy of approaches together with a brief description of their main features. In the final part of this paper, we also identify what we believe to be promising areas of future research in the field of CMOEAs.

Recently, it was found that most multiobjective particle swarm optimizers (MOPSOs) perform poorly when tackling many-objective optimization problems (MaOPs). This is mainly because the loss of selection pressure that occurs when updating the swarm. The number of nondominated individuals is substantially increased and the diversity maintenance mechanisms in MOPSOs always guide the particles to explore sparse regions of the search space. This behavior results in the final solutions being distributed loosely in objective space, but far away from the true Pareto-optimal front. To avoid the above scenario, this paper presents a balanceable fitness estimation method and a novel velocity update equation, to compose a novel MOPSO (NMPSO), which is shown to be more effective to tackle MaOPs. Moreover, an evolutionary search is further run on the external archive in order to provide another search pattern for evolution. The DTLZ and WFG test suites with 4-10 objectives are used to assess the performance of NMPSO. Our experiments indicate that NMPSO has superior performance over four current MOPSOs, and over four competitive multiobjective evolutionary algorithms (SPEA2-SDE, NSGA-III, MOEA/DD, and SRA), when solving most of the test problems adopted.

Hyper-heuristics are powerful search methodologies that can adapt to different kinds of problems. One element of paramount importance, however, is the selection module that they incorporate. Traditional approaches define a set of features for characterizing a problem and, thus, define how to best solve it. However, some features may vary nonlinearly as the solver progresses, requiring higher resolution in specific areas of the feature domain. This work focuses on assessing the advantage of using feature transformations to improve the given resolution and, as a consequence, to improve the overall performance of a hyper-heuristic. We provide evidence that using feature transformations may result in a better discrimination of the problem instance and, as consequence, a better performance of the hyper-heuristics. The feature transformation strategy was applied to an evolutionary-based hyper-heuristic model taken from the literature and tested on constraint satisfaction problems The proposed strategy increased the median success rate of hyper-heuristics by more than 13% and reduced its standard deviation in about 7%, while reducing the median number of adjusted consistency checks by almost 30%.

Hyper-heuristics have emerged as an important strategy for combining the strengths of different heuristics into a single method. Although hyper-heuristics have been found to be successful in many scenarios, little attention has been paid to the subsets of heuristics that these methods manage and apply. In several cases, heuristics can interfere with each other and can be harmful for the search. Thus, obtaining information about the differences among heuristics, and how they contribute to the search process is very important. The main contribution of this paper is an automatic heuristic-filtering process that allows hyper-heuristics to exclude heuristics that do not contribute to improving the solution. Based on some previous works in feature selection, two methods are proposed that rank heuristics and sequentially select only suitable heuristics in a hyper-heuristic framework. Our experiments over a set of Constraint Satisfaction Problem instances show that a hyper-heuristic with only selected heuristics obtains significantly better results than a hyper-heuristic containing all heuristics, in terms of running times. In addition, the success rate of solving such instances is better for the hyper-heuristic with the suitable heuristics than for the hyper-heuristic without our proposed filtering process.

In recent years, the design of new selection mechanisms based on quality indicators has become a popular trend in the development of Multi-Objective Evolutionary Algorithms (MOEAs). This trend has been motivated by the well-known limitations of Pareto-based MOEAs when dealing with many-objective optimization problems (i.e., problems having more than 3 objectives). In this paper, we propose a selection mechanism (called IGD+-H) which is based on the combination of the Inverted Generational Distance+(IGD+) indicator and Kuhn-Munkres' (Hungarian) algorithm to solve Linear Assignment Problems (LAPs). The proposed selection scheme is compared with respect to other selection mechanisms based on the IGD indicator and with respect to the use of the Δpindicator. Our proposed technique is incorporated into a MOEA and is validated using standard test functions. Our comparative study indicates that both Δpand IGD present some limitations when selecting solutions in degenerate multi-objective problems. Our results show that the transformation of the selection mechanism into a linear assignment problem speeds up the convergence of the MOEA and it is able to solve many-objective problems in an effective and efficient manner. We show that our proposed IGD+-H-based selection mechanism is able to achieve a significant speed up (of up to 200×) with respect to the exclusive use of any of the indicators adopted in our study.

Nowadays, heuristics represent a commonly used alternative to solve complex optimization problems. This, however, has given rise to the problem of choosing the most effective heuristic for a given problem. In recent years, one of the most used strategies for this task has been the hyper-heuristics, which aim at selecting/generating heuristics to solve a wide range of optimization problems. Most of the existing selection hyper-heuristics attempt to recommend only one heuristic for a given instance. However, for some classes of problems, more than one heuristic can be suitable. With this premise, in this paper, we address this issue through an evolutionary multilabel learning approach for building hyper-heuristics. Unlike traditional approaches, in the multilabel formulation, the result could not be a single recommendation, but a set of potential heuristics. Due to the fact that cooperative coevolutionary algorithms allow us to divide the problem into several subproblems, it results in a natural approach for dealing with multilabel classification. The proposed cooperative coevolutionarymultilabel approach aims at choosing the most relevant patterns for each heuristic. For the experimental study included in this paper, we have used a set of constraint satisfaction problems as our study case. Our experimental results suggest that the proposed method is able to generate accurate hyper-heuristics that outperform reference methods.

Scalarizing functions have been successfully used by Multi-Objective Evolutionary Algorithms (MOEAs) for the fitness assignment process. Their popularity has to do with their low computational cost, their capability to generate (weakly) Pareto optimal solutions, and their effectiveness in solving many-objective optimization problems. Nevertheless, recent studies indicate that the search behavior of MOEAs strongly depends on the choice of the scalarizing function. Besides, this specification varies according to the Pareto-front geometry of the problem at hand. In this work, we present a novel hyper-heuristic for continuous search spaces, which combines the strengths and compensates for the weaknesses of different scalarizing functions. These heuristics have been proposed within the evolutionary multi-objective optimization and mathematical programming communities. Furthermore, the selection of heuristics is conducted through the s-energy, which measures the even distribution of a set of points in k-dimensional manifolds. Experimental results indicate that our proposed approach outperforms the use of a single heuristic as well as other state-of-the-art algorithms in the majority of the ZDT, DTLZ and WFG test problems.

It is well known that the performances of evolutionary algorithms are influenced by the quality of their initial populations. Over the years, many different techniques for generating an initial population by uniformly covering as much of the search space as possible have been proposed. However, none of these approaches considers any input from the function that must be evolved using that population. In this paper, a new initialization technique, which can be considered a heuristic space-filling approach, based on both function to be optimized and search space, is proposed. It was tested on two well-known unconstrained sets of benchmark problems using several computational intelligence algorithms. The results obtained reflected its benefits as the performances of all these algorithms were significantly improved compared with those of the same algorithms with currently available initialization techniques. The new technique also proved its capability to provide useful information about the function's behavior and, for some test problems, the initial population produced high-quality solutions. This method was also tested on a few multiobjective problems, with the results demonstrating its benefits.

The selection of swarm leaders (i.e., the personal best and global best), is important in the design of a multiobjective particle swarm optimization (MOPSO) algorithm. Such leaders are expected to effectively guide the swarm to approach the true Pareto optimal front. In this paper, we present a novel external archive-guided MOPSO algorithm (AgMOPSO), where the leaders for velocity update are all selected from the external archive. In our algorithm, multiobjective optimization problems (MOPs) are transformed into a set of subproblems using a decomposition approach, and then each particle is assigned accordingly to optimize each subproblem. A novel archive-guided velocity update method is designed to guide the swarm for exploration, and the external archive is also evolved using an immune-based evolutionary strategy. These proposed approaches speed up the convergence of AgMOPSO. The experimental results fully demonstrate the superiority of our proposed AgMOPSO in solving most of the test problems adopted, in terms of two commonly used performance measures. Moreover, the effectiveness of our proposed archive-guided velocity update method and immune-based evolutionary strategy is also experimentally validated on more than 30 test MOPs.

Support vector machines (SVMs) are among the most powerful learning algorithms for classification tasks. However, these algorithms require a high computational cost during the training phase, which can limit their application on large-scale datasets. Moreover, it is known that their effectiveness highly depends on the hyper-parameters used to train the model. With the intention of dealing with these, this paper introduces an evolutionary multiobjective model and instance selection (IS) approach for SVMs with Pareto-based ensemble, whose goals are, precisely, to optimize the size of the training set and the classification performance attained by the selection of the instances, which can be done using either a wrapper or a filter approach. Due to the nature of multiobjective evolutionary algorithms, several Pareto optimal solutions can be found. We study several ways of using such information to perform a classification task. To accomplish this, our proposal performs a processing over the Pareto solutions in order to combine them into a single ensemble. This is done in five different ways, which are based on: 1) a global Pareto ensemble; 2) error reduction; 3) a complementary error reduction; 4) maximized margin distance; and 5) boosting. Through a comprehensive experimental study we evaluate the suitability of the proposed approach and the Pareto processing, and we show its advantages over a single-objective formulation, traditional IS techniques, and learning algorithms.

Cooperative coevolutionary algorithms (CCAs) are extensions of traditional Evolutionary Algorithms (EAs) that have a lot of potential in addressing some problems on which EAs tend to perform poorly. CCAs have become an important area of research within evolutionary computation and since the cooperative coevolutionary framework was extended to multi-objective optimization, a number of approaches have been proposed incorporating it as a means for improving the performance of multi-objective EAs. The advantage of CCAs is the decomposition of the problem they use, which allows us to learn different parts of the problem instead of the whole problem at once. Cooperative coevolution has a symbiotic approach that evolves species populations (each one managing a part of the problem) which are evaluated based on how well they perform together. In order to form a solution, an individual from each species is selected and combined with the other selected individuals. The solution is then evaluated and the individuals that make up the solution are scored based on the fitness of the combined solution. The way this selection to collaborate is done is a key issue in a cooperative coevolutionary framework. However, the usual approach that has been used in Cooperative Coevolutionary Multi-objective EAs (CCMOEAs) is a method based on Pareto optimality. In this work, we present a novel collaboration formation mechanism for CCMOEAs based on the use of the hypervolume indicator. Our preliminary results confirm the impact that the collaboration mechanism has on the performance of CCMOEAs and indicate that our proposed framework clearly improves the results obtained by a CCMOEA whose selection mechanism for collaboration is based on Pareto optimality.

Over the last two decades, many differential evolution algorithms have been introduced to solve constrained optimization problems. Due to the variability of characteristics of such problems, no single algorithm performs consistently well over all of them. In this paper, for a better coverage of the problem characteristics, we introduce an enhanced multi-operator differential evolution algorithm, which utilizes the strengths of multiple search operators at each generation, and places more emphasis on the best-performing ones during the optimization process based on three measures: (1) the quality of solutions; (2) the feasibility rate; and (3) diversity. In addition, an improved self-adaptive mechanism for automatically controlling the scaling factor and crossover rate is proposed. The performance of the algorithm is assessed using a well-known set of constrained problems, with the experimental results demonstrating that it is superior to state-of-the-art algorithms.

In this paper, an Exponential Weighting Moving Average (EWMA) control parameter adaptation technique is tested with Generalized Differential Evolution 3 (GDE3) using a set of multi-objective test problems and performance metrics. The results with and without EWMA control parameter adaptation are compared. EWMA has been earlier proposed with the original unconstrained single-objective Differential Evolution (DE), and EWMA adapts crossover and mutation control parameter values. From the results, it is observed that if good initial control parameter values are used, then there is not clear performance difference between the original GDE3 and GDE3 with EWMA. However, if the initial control parameter values are not good, then EWMA gives clear improvement in performance. Since GDE3 with EWMA is identical with the original DE in the case of single-objective optimization, the same control parameter adaptation technique can be used both in the case of single- and multi-objective optimization. However, different initial control parameter values should be used in different cases, and recommendations for initial values are given at the end of the paper.

In recent years, the design of selection mechanisms based on performance indicators has become a very popular trend in the development of new Multi-Objective Evolutionary Algorithms (MOEAs). The main motivation has been the well-known limitations of Pareto-based MOEAs when dealing with problems having four or more objectives (the so-called many-objective problems). The most commonly adopted indicator has been the hypervolume, mainly because of its nice mathematical properties (e.g., it is the only unary indicator which is known to be Pareto compliant). However, the hypervolume has a well-known disadvantage: its exact computation is very costly in high dimensionality, making it prohibitive for many-objective problems (this cost normally becomes unaffordable for problems with more than 5 objectives). Recently, a variation of the well-known inverse generational distance (IGD) was introduced. This indicator, which is called IGD+was shown to be weakly Pareto compliant, and presents some evident advantages with respect to the original IGD. Here, we propose an indicator-based MOEA, which adopts IGD+. The proposed approach adopts a novel technique for building the reference set, which is used to assess the quality of the solutions obtained during the search. Our preliminary results indicate that our proposed approach is able to solve many-objective problems in an effective and efficient manner, being able to obtain solutions of a similar quality to those obtained by SMS-EMOA and MOEA/D, but at a much lower computational cost than required by the computation of exact hypervolume contributions (as adopted in SMS-EMOA).

In this paper, we propose a new selection scheme for Multi-Objective Evolutionary Algorithms (MOEAs) based on the Δρindicator. Our new selection scheme is incorporated into a MOEA giving rise to the “Δρ-MOEA.” Perhaps, one of the most important disadvantages of MOEAs based on Δρis the definition of the reference set. In this work, we propose to create a reference set at each generation using e-dominance and the set of nondominated solutions found so far. Our new selection scheme uses two different techniques to select solutions according to the modified generational distance indicator or the modified inverted generational distance indicator. Our proposed Δp-MOEA is validated using standard test functions taken from the specialized literature, having three to six objective functions and it is compared with respect to two well-known MOEAs: MOEA/D using Penalty Boundary Intersection (PBI), which is based on decomposition, and SMS-EMOA-HYPE (a version of SMS-EMOA that uses a fitness assignment scheme based on the use of an approximation of the hypervolume indicator).

Multi-Objective Evolutionary Algorithms (MOEAs) are powerful tools for solving a wide range of real-world applications that involve the simultaneous optimization of several objective functions. However, their scalability to many-objective problems remains as an important issue since, due to the large number of non-dominated solutions, the search is guided solely by the diversity criterion. In this paper, we propose a novel MOEA that incorporates a density estimator based on a visualization technique called Parallel Coordinates. Using this approach, a graph is represented by a digital image, where a pixel identifies the level of overlapping line segments and those individuals covering a wide area of the image have a high probability of survival. Experimental results indicate that our proposed approach, called Multi-objective Optimizer based on Value Path (MOVAP), outperforms existing algorithms based on clustering (SPEA2), crowding distance (NSGA-II), reference points (NSGA-III) and the hypervolume indicator (HypE) on most of the problems of the WFG test suite for five and seven objectives, while its performance in low dimensionality remains competitive.

The design of bar structures in civil engineering is a complex problem when dealing with real-world structures. An approach to deal with these problems is to apply metaheuristics, which are stochastic methods based on iteratively producing and evaluating tentative solutions. In particular, we focus on multi-objective metaheuristics, as we consider two goals to be minimized: the weight and the deflection of the structure. When applying these techniques to real-world problems, running a metaheuristic for several thousands of evaluations may require many days on a single processor. In this paper, we develop distributed master/slave versions of four multi-objective metaheuristics that are representative of the state-of-the-art and apply them to optimize the design of two instances of a cable-strayed bridge. Our study reveals that our parallel proposals are able to effectively use up to 450 cores, providing accurate solutions in a short time.

k-NN is one of the most popular and effective classifiers nowadays. However, it has some limitations that overcome its applicability in large scale scenarios: basically, it requires storing the whole training set, and it computes distances of a test sample with the training data set. These limitati ons have been traditionally alleviated with data reduction techniques. This paper introduces a multi-objective evolutionary approach for data reduction. Our method simultaneously generates prototypes and selects features for k-NN classifiers. Contrary to most of the existing approaches, our method treats the problem with multi-objective evolutionary optimizers. We show the effectiveness of our proposal in benchmark data and compare its performance with state of the art techniques.

Premature convergence is one of the best-known drawbacks that affects the performance of evolutionary algorithms. An alternative for dealing with this problem is to explicitly try to maintain proper diversity. In this paper, a new replacement strategy that preserves useful diversity is presented. The novelty of our method is that it combines the idea of transforming a single-objective problem into a multiobjective one, by considering diversity as an explicit objective, with the idea of adapting the balance induced between exploration and exploitation to the various optimization stages. Specifically, in the initial phases, larger amounts of diversity are accepted. The diversity measure considered in this paper is based on calculating distances to the closest surviving individual. Analyses with a multimodal function better justify the design decisions and provide greater insight into the working operation of the proposal. Computational results with a packing problem that was proposed in a popular contest illustrate the usefulness of the proposal. The new method significantly improves on the best results known to date for this problem and compares favorably against a large number of state-of-the-art schemes.

In recent years, multiobjective immune algorithms (MOIAs) have shown promising performance in solving multiobjective optimization problems (MOPs). However, basic MOIAs only use a single hypermutation operation to evolve individuals, which may induce some difficulties in tackling complicated MOPs. In this paper, we propose a novel hybrid evolutionary framework for MOIAs, in which the cloned individuals are divided into several subpopulations and then evolved using different evolutionary strategies. An example of this hybrid framework is implemented, in which simulated binary crossover and differential evolution with polynomial mutation are adopted. A fine-grained selection mechanism and a novel elitism sharing strategy are also adopted for performance enhancement. Various comparative experiments are conducted on 28 test MOPs and our empirical results validate the effectiveness and competitiveness of our proposed algorithm in solving MOPs of different types.

Cooperative coevolution is an approach for evolving solutions from different populations which are evaluated based on how well they perform together. The advantage of cooperative coevolutionary algorithms is the decomposition of the problem which allows us to learn different parts of the problem instead of the whole problem at once. However, previous research within the field of global optimization has shown that cooperative coevolutionary algorithms are biased towards equilibrium states. Since studies concerning cooperative coevolutionary algorithms used for solving multi-objective optimization problems were initiated, no attention has been paid to this issue. In this paper, we show empirical evidence of the existence of these problems within the multi-objective optimization field and present a novel cooperative coevolution framework which, through the use of the concept of Nash equilibrium, alleviates some of those optimization-related pathologies present in cooperative coevolutionary algorithms. We compare our proposed algorithm with respect to two algorithms that make use of the cooperative coevolutionary model to multi-objective optimization, NSCCGA (that makes use of Potter's coevolutionary model) and GCEA (a game theory based coevo-lutionary algorithm). The computational effort required by each algorithm (measured in terms of the number of fitness function evaluations) is also analyzed. Our preliminary results indicate that the proposed framework clearly outperforms the results of the aforementioned algorithms when using the Deb-Thiele-Laumanns-Zitzler (DTLZ) and the Zitzler-Deb-Thiele (ZDT) test suites.

In spite of the success of the multi-objective evolutionary algorithm based on decomposition (MOEA/D), the generation of weights for problems having many objectives, continues to be an open research problem. In this paper, we introduce a new methodology based on low-discrepancy sequences to generate the weights vectors employed by MOEA/D. We analyze and compare the proposed methodology using different low-discrepancy sequences and its impact in the search process of MOEA/D. The proposed approach is evaluated in problems having many objective functions (up to 15 objectives). We show the flexibility and ease of use of this type of sequences when adopting them to generate the weights of MOEA/D.

In this paper, we propose a new selection mechanism based on ε-dominance which is called “ε-selection”. An interesting feature of this selection scheme is that it does not require to set the value of ε ahead of time. Our ε-selection is incorporated into the GD-MOEA algorithm, giving rise to the so-called “Generational Distance & ε-dominance Multi-Objective Evolutionary Algorithm (GDE-MOEA)”. Our proposed GDE-MOEA is validated using standard test functions taken from the specialized literature, having three to six objective functions. GDE-MOEA is compared with respect to the original GD-MOEA, which is based on the generational distance indicator and a technique based on Euclidean distances to improve the diversity in the population. Additionally, our proposed approach is compared with respect to MOEA/D using Penalty Boundary Intersection (PBI), which is based on decomposition, and SMS-EMOA-HYPE (a version of SMS-EMOA that uses a fitness assignment scheme based on the use of an approximation of the hypervolume indicator). Our preliminary results indicate that our proposed GDE-MOEA is a good alternative to solve multi-objective optimization problems having both low dimensionality and high dimensionality in objective function space because it obtains better results than GD-MOEA and MOEA/D in most cases and it is competitive with respect to SMS-EMOA-HYPE but at a much lower computational cost.

Particle swarm optimization (PSO) algorithms have been widely used to solve a variety of optimization problems. Their success has motivated researchers to extend the use of these techniques to the multi-objective optimization field. However, most of these extensions have been used to solve multi-objective optimization problems (MOPs) with no more than three objective functions. Here, we propose a novel multi-objective PSO (MOPSO) algorithm characterized by the use of a recent approach that transforms a MOP into a linear assignment problem (LAP), with the aim of being able to solve many-objective optimization problems. Our proposed approach, called LAP based PSO (LAPSO), adopts the Munkres assignment algorithm to solve the generated LAPs and has no need of an external archive. LAPSO is compared with respect to three MOPSOs which are representative of the state-of-the-art in the area: the Optimized Multi-Objective Particle Swarm Optimizer (OMOPSO) the Speed-constrained Multiobjective Particle Swarm Optimizer (SMPSO) and a variant of the latter that uses the hypervolume indicator for its leader selection scheme (SMPSOhv). Our results indicate that LAPSO is able to outperform the MOPSOs with respect to which it was compared in most of the test problems adopted, specially when solving instances with more than three objectives.

In recent years, performance indicators were introduced as a selection mechanism in multi-objective evolutionary algorithms (MOEAs). A very attractive option is the R2 indicator due to its low computational cost and weak-Pareto compatibility. This indicator requires a set of utility functions, which map each objective to a single value. However, not all the utility functions available in the literature scale properly for more than four objectives and the diversity of the approximation sets is sensitive to the choice of the reference points during normalization. In this paper, we present an improved version of a MOEA based on the $R2$ indicator, which takes into account these two key aspects, using the achievement scalarizing function and statistical information about the population's proximity to the true Pareto optimal front. Moreover, we present a comparative study with respect to some other emerging approaches, such as NSGA-III (based on Pareto dominance), Δp-DDE (based on the Δp indicator) and some other MOEAs based on the R2 indicator, using the DTLZ and WFG test problems. Experimental results indicate that our approach outperforms the original algorithm as well as the other MOEAs in the majority of the test instances, making it a suitable alternative for solving many-objective optimization problems.

Time varying nature of the constraints, objectives and parameters that characterize several practical optimization problems have led to the field of dynamic optimization with Evolutionary Algorithms. In recent past, very few researchers have concentrated their efforts on the study of Dynamic multi-objective Optimization Problems (DMOPs) where the dynam-icity is attributed to multiple objectives of conflicting nature. Considering the lack of a somewhat diverse and challenging set of benchmark functions, in this article, we discuss some ways of designing DMOPs and propose some general techniques for introducing dynamicity in the Pareto Set and in the Pareto Front through shifting, shape variation, slope variation, phase variation, and several other types. We introduce 9 benchmark functions derived from the benchmark suite used for the 2009 IEEE Congress on Evolutionary Computation competition on bound-constrained and static MO optimization algorithms. Additionally a variant of multiobjective EA based on decomposition (MOEA/D) have been put forward and tested along with peer algorithms to evaluate the newly proposed benchmarks.

This paper proposes a new hypervolume-based multi-objective particle swarm optimizer (called MOPSOhv) that uses an external archive to store the global nondominated solutions found during the evolutionary process. The proposed algorithm makes use of the hypervolume contribution of archived solutions for selecting global and personal leaders for each particle in the main swarm, and also as a mechanism for pruning the external archive when it is updated with new nondominated solutions. In order to increase the diversity when particles are updated in their positions, a mutation operator is used. The performance of the proposed algorithm is evaluated adopting standard test problems and indicators reported in the specialized literature, comparing its results with respect to those obtained by state-of-the-art multi-objective evolutionary algorithms. Our preliminary results indicate that our proposal is competitive with respect to state-of-the-art multi-objective evolutionary algorithms, being particularly suitable for solving many-objective optimization problems (i.e., problems having more than 3 objectives).

In spite of the popularity of the Multi-objective Evolutionary Algorithm based on Decomposition (MOEA/D), its use in Constrained Multi-objective Optimization Problems (CMOPs) has not been fully explored. In the last few years, there have been a few proposals to extend MOEA/D to the solution of CMOPs. However, most of these proposals have adopted selection mechanisms based on penalty functions. In this paper, we present a novel selection mechanism based on the well-known ε-constraint method. The proposed approach uses information related to the neighborhood adopted in MOEA/D in order to obtain solutions which minimize the objective functions within the allowed feasible region. Our preliminary results indicate that our approach is highly competitive with respect to a state-of-the-art MOEA which solves in an efficient way the constrained test problems adopted in our comparative study.

In this paper, we propose a new selection mechanism based on the maximin fitness function and a technique based on Euclidean distances between solutions to improve the diversity of the population in objective function space. Our new selection mechanism is incorporated into a multi-objective evolutionary algorithm (MOEA) which uses the operators of NSGA-II (crossover and mutation) to generate new individuals, giving rise to the so-called “Maximin-Distances Multi-Objective Evolutionary Algorithm (MD-MOEA)”. Our MD-MOEA is validated using standard test functions taken from the specialized literature, having three to six objective functions. MD-MOEA is compared with respect to MC-MOEA (which is based on the maximin fitness function and a clustering technique), MOEA/D using Penalty Boundary Intersection (PBI), which is based on decomposition, and SMS-EMOA-HYPE (a version of SMS-EMOA that uses a fitness assignment based on the use of an approximation of the hypervolume indicator). Our preliminary results indicate that our MD-MOEA is a good alternative to solve multi-objective optimization problems having both low dimensionality and high dimensionality in objective function space because it obtains better results than MC-MOEA and MOEA/D in most cases and it is competitive with respect to SMS-EMOA-HYPE (in fact, it outperforms SMS-EMOA-HYPE in problems of high dimensionality) but at a much lower computational cost.

k-NN is one of the most popular and effective models for pattern classification. However, it has two main drawbacks that hinder the application of this method for large data sets: (1) the whole training set has to be stored in memory, and (2) for classifying a test pattern it has to be compared to all other training instances. In order to overcome these shortcomings, prototype generation (PG) methods aim to reduce the size of the training set while maintaining or increasing the classification performance of k-NN. Accordingly, most PG methods aim to generate instances that try to maximize classification performance. Nevertheless, in most cases, the reduction objective is only implicitly optimized. This paper introduces EMOPG, a novel approach to PG based on multi-objective optimization that explicitly optimizes both objectives: accuracy and reduction. Under EMOPG, prototypes are initialized with a subset of training instances selected through a tournament, according to a weighting term. A multi-objective evolutionary algorithm, PAES (Pareto Archived Evolution Strategy), is implemented to adjust the position of the initial prototypes. The optimization process aims to simultaneously maximize the classification performance of prototypes while reducing the number of instances with respect to the training set. A strategy for selecting a single solution from the set of non-dominated solutions is proposed. We evaluate the performance of EMOPG using a suite of benchmark data sets and compare the performance of our proposal with respect to the one obtained by alternative techniques. Experimental results show that our proposed method offers a better trade-off between accuracy and reduction than other methods.

Differential Evolution (DE) is a very efficient meta-heuristic for optimization over continuous spaces which has gained much popularity in recent years. Several parameter control strategies have been proposed to automatically adapt its internal parameters. The most advanced DE variants take into account the feedback obtained in the optimization process to guide the dynamic setting of the DE parameters. Indeed, the automatic adaptation of the crossover rate (CR) has attracted a lot of research in the last decades. In most of such strategies, the quality of using a given CR value is measured by considering the probability of performing a replacement in the DE selection stage when such a value is applied. One of the main contributions of this paper is to experimentally show that the probability of replacement induced by the application of a given CR value and the quality of the obtained results are not as correlated as expected. This might cause a performance deterioration that avoids the achievement of good quality solutions even in the long-term. In addition, the experimental evaluation developed with a set of optimization problems of varying complexities clarifies some of the advantages and drawbacks of the different tested strategies. The only component varied among the different tested schemes has been the CR control strategy. The study presented in this paper provides advances in the understanding of the inner working of several state-of-the-art adaptive DE variants.

In this paper, we provide a general introduction to the so-called multi-objective evolutionary algorithms, which are metaheuristic search techniques inspired on natural evolution that are able to deal with highly complex optimization problems having two or more objectives. In the first part of the paper, we provide some basic concepts necessary to make the paper self-contained, as well as a short review of the most representative multi-objective evolutionary algorithms currently available in the specialized literature. After that, a short review of applications of these algorithms in pattern recognition is provided. The final part of the paper presents some possible future research paths in this area as well as our conclusions.

In this paper, we present a Multi-objective Particle Swarm Optimizer (MOPSO) based on a decomposition approach, which is proposed to solve Constrained Multi-Objective Aerodynamic Shape Optimization Problems (CMO-ASOPs). The constraint-handling technique adopted in this approach is based on the well-known epsilon-constraint method. Since the ε-constraint method was initially proposed to deal with constrained single-objective optimization Problems, we adapted it so that it could be incorporated into a MOPSO. Our main focus is to solve CMO-ASOPs in an efficient and effective manner. The proposed constrained MOPSO guides the search by updating the position of each particle using a set of solutions considered as the global best according to both the decomposition approach and the epsilon-constraint method. Our preliminary results indicate that our proposed approach is able to outperform a state-of-the-art MOEA in several CMO-ASOPs.

The aim of any data mining technique is to build an efficient predictive or descriptive model of a large amount of data. Applications of evolutionary algorithms have been found to be particularly useful for automatic processing of large quantities of raw noisy data for optimal parameter setting and to discover significant and meaningful information. Many real-life data mining problems involve multiple conflicting measures of performance, or objectives, which need to be optimized simultaneously. Under this context, multiobjective evolutionary algorithms are gradually finding more and more applications in the domain of data mining since the beginning of the last decade. In this two-part paper, we have made a comprehensive survey on the recent developments of multiobjective evolutionary algorithms for data mining problems. In this paper, Part I, some basic concepts related to multiobjective optimization and data mining are provided. Subsequently, various multiobjective evolutionary approaches for two major data mining tasks, namely feature selection and classification, are surveyed. In Part II of this paper, we have surveyed different multiobjective evolutionary algorithms for clustering, association rule mining, and several other data mining tasks, and provided a general discussion on the scopes for future research in this domain.

This paper is the second part of a two-part paper, which is a survey of multiobjective evolutionary algorithms for data mining problems. In Part I , multiobjective evolutionary algorithms used for feature selection and classification have been reviewed. In this part, different multiobjective evolutionary algorithms used for clustering, association rule mining, and other data mining tasks are surveyed. Moreover, a general discussion is provided along with scopes for future research in the domain of multiobjective evolutionary algorithms for data mining.

The six articles in this special issue provide a snapshot of the current research trends in multiobjective evolutionary algorithms for data mining. The main issues and challenges in this domain have been highlighted and directions of future research work have also been provided,

Many real-world multi-objective optimization problems have hundreds or even thousands of decision variables, which contrast with the current practice of multi-objective metaheuristics whose performance is typically assessed using benchmark problems with a relatively low number of decision variables (normally, no more than 30). In this paper, we propose a cooperative coevolution framework that is capable of optimizing large scale (in decision variable space) multi-objective optimization problems. We adopt a benchmark that is scalable in the number of decision variables (the ZDT test suite) and compare our proposed algorithm with respect to two state-of-the-art multi-objective evolutionary algorithms (GDE3 and NSGA-II) when using a large number of decision variables (from 200 up to 5000). The results clearly indicate that our proposed approach is effective as well as efficient for solving large scale multi-objective optimization problems.

In recent years, the development of selection mechanisms based on performance indicators has become an important trend in algorithmic design. Hereof, the hypervolume has been the most popular choice. Multi-objective evolutionary algorithms (MOEAs) based on this indicator seem to be a good choice for dealing with many-objective optimization problems. However, their main drawback is that such algorithms are typically computationally expensive. This has motivated some recent research in which the use of other performance indicators has been explored. Here, we propose an efficient mechanism to integrate the R2 indicator to a modified version of Goldberg's nondominated sorting method, in order to rank the individuals of a MOEA. Our proposed ranking scheme is coupled to two different search engines, resulting in two new MOEAs. These MOEAs are validated using several test problems and performance measures commonly adopted in the specialized literature. Results indicate that the proposed ranking approach gives rise to effective MOEAs, which produce results that are competitive with respect to those obtained by three well-known MOEAs. Additionally, we validate our resulting MOEAs in many-objective optimization problems, in which our proposed ranking scheme shows its main advantage, since it is able to outperform a hypervolume-based MOEA, requiring a much lower computational time.

The incorporation of performance indicators as the selection mechanism of a multi-objective evolutionary algorithm (MOEA) is a topic that has attracted increasing interest in the last few years. This has been mainly motivated by the fact that Pareto-based selection schemes do not perform properly when solving problems with four or more objectives. The indicator that has been most commonly used for being incorporated in the selection mechanism of a MOEA has been the hypervolume. Here, however, we explore the use of the R2 indicator, which presents some advantages with respect to the hypervolume, the main one being its low computational cost. In this paper, we propose a new MOEA called Many-Objective Metaheuristic Based on the R2 Indicator (MOMBI), which ranks individuals using a utility function. The proposed approach is compared with respect to MOEA/D (based on scalarization) and SMS-EMOA (based on hypervolume) using several benchmark problems. Our preliminary experimental results indicate that MOMBI obtains results of similar quality to those produced by SMS-EMOA, but at a much lower computational cost. Additionally, MOMBI outperforms MOEA/D in most of the test instances adopted, particularly when dealing with high-dimensional problems having complicated Pareto fronts. Thus, we believe that our proposed approach is a viable alternative for solving many-objective optimization problems.

This paper presents the goal-constraint method for incorporating preferences in multiobjective optimization. The preferences are provided in the form of a vector of goals, which is familiar for decision makers and operations researchers. The portion of the Pareto front to be generated is totally defined by the vector of goals, regardless if such a vector is feasible or not. Once defined, it is feasible to experiment on many objective problems, because of the reduced cost of producing less points. The experimental results show good convergence properties, and the graphs illustrate the way the portion of front produced is related to the vector of goals.

The development of multi-objective evolutionary algorithms (MOEAs) assisted by surrogate models has significantly increased in the last few years. However, in realworld applications, the high modality and dimensionality that functions normally have, often causes problems to such models. Therefore, if the Pareto optimal set of a multi-objective optimization problem is located in a search space in which the surrogate model is not able to shape the corresponding region, the search could be misinformed and thus converge to wrong regions. This has motivated the idea of incorporating refinement mechanisms to such approaches, in order to improve the search. In this paper, we present a local search mechanism which improves the search of a MOEA assisted by surrogate models. Our preliminary results indicate that our proposed approach can produce good quality results when it is restricted to performing only between 1,000 and 5,000 fitness function evaluations. Our proposed approach is validated using a set of standard test problems and an airfoil design problem.

In this paper, we propose a new selection mechanism based on the hypervolume indicator and on its “locality property”, which is incorporated into the SMSEMOA, giving rise to the so-called improved SMS-EMOA (iSMS-EMOA). Our proposed selection mechanism is validated using standard test functions taken from the specialized literature, having three to six objective functions. iSMS-EMOA is compared with respect to its predecessor SMS-EMOA and with respect to another version of SMS-EMOA that uses the approximation of the hypervolume indicator, instead of its exact calculation. Our preliminary results indicate that our proposed selection mechanism outperforms the selection mechanisms based on the hypervolume indicator that have been proposed in recent years, since it significantly reduces the computational time required by the algorithm without sacrificing quality in the approximation generated.

Algorithms based on the Particle Swarm Optimization (PSO) scheme have become popular to solve both single- and multi-objective optimization problems. In this paper, we focus on SMPSO, a PSO designed to cope with this second group of problems. Taking it as our starting point, we analyze different leader selection schemes, which give rise to four new variants of SMPSO. These new versions, along with the original algorithm, are compared using a benchmark composed of 21 problems. Our study reveals that SMPSOhv, a variant that uses the hypervolume indicator to guide leader selection, is the best performing algorithm in our comparison, outperforming also the original version of SMPSO. To further assess the performance of SMPSOhv, we compare it against NSGA-II and SMS-EMOA, achieving again the best overall results in this new comparative study. Based on these observations, we conclude that the use of the hypervolume for leader selection is a promising approach for multi-objective PSO algorithms.

Dynamic Constrained Optimization Problems (DCOP) are a unique class of optimization problems where the objective function as well as the constraint functions change with respect to time. Conventional DCO algorithms involve Genetic Algorithms (GAs) accompanied by a separate constraint-handling technique e.g., a repair method, or a penalty function. However, ordinary repair methods with elitism significantly decrease the diversity of the population during the exploitation stage and penalty functions cannot properly deal with disconnected feasible regions. In this paper, we propose a new approach based on the Gravitational Search Algorithm as well as a modified version of a repair method that produces improved results. The proposed approach incorporates knowledge-reusing and knowledge-restarting in order to produce a quick recovery and faster convergence.

Evolutionary algorithms have gained popularity as an alternative for dealing with multi-objective optimization problems. However, these algorithms require to perform a relatively high number of fitness function evaluations in order to generate a reasonably good approximation of the Pareto front. This can be a shortcoming when fitness evaluations are computationally expensive. In this paper, we propose an approach that combines an evolutionary algorithm with an ensemble of surrogate models based on support vector machines (SVM), which are used to approximate the fitness functions of a problem. The proposed approach performs a model selection process for determining the appropriate hyperparameters values for each SVM in the ensemble. The ensemble is constructed in an incremental fashion, such that the models are updated with the knowledge gained during the evolutionary process, but the information from previous evaluated regions is also preserved. A criterion based on surrogate fidelity is also proposed for determining when should the surrogates be updated. We evaluate the performance of our proposal using a benchmark of test problems widely used in the literature and we compare our results with respect to those obtained by the NSGA-II. Our proposed approach is able to significantly reduce the number of fitness function evaluations performed, while producing solutions which are close to the true Pareto front.

The maintenance of a proper diversity is an important issue for the correct behavior of Evolutionary Algorithms (EAs). The loss of diversity might lead to stagnation in suboptimal regions, producing the effect known as “premature convergence”. Several methods to avoid premature convergence have been previously proposed. Among them, the use of Multi-objective Evolutionary Algorithms (MOEAs) is a promising approach. Several ways of using MOEAs for single-objective optimization problems have been devised. The use of an additional objective based on calculating the diversity that each individual introduces in the population has been successfully applied by several researchers. Several ways of measuring the diversity have also been tested. In this work, the main weaknesses of some of the previously presented approaches are analyzed. Considering such drawbacks, a new scheme whose aim is to maintain a better diversity than previous approaches is proposed. The proposed approach is empirically validated using a set of well-known single-objective benchmark problems. Our preliminary results indicate that the proposed approach provides several advantages in terms of premature convergence avoidance. An analysis of the convergence in the average-case is also carried out. Such an analysis reveals that the better ability of our proposed approach to deal with premature convergence produces a reduction in the convergence speed in the average-case for several of the benchmark problems adopted.

This paper employs an evolutionary algorithm to adjust the weights of the evaluation function of a chess engine. The selection mechanism of this algorithm chooses the virtual players (individuals in the population) that have the highest number of problems properly solved from a database of tactical and positional chess problems. This method has as its main advantage that we only mutate those weights involved in the solution of the current problem. Furthermore, the mutation mechanism is based on a Gaussian distribution whose standard deviation is adapted through the number of problems solved by each virtual player. We show here how, with the use of this method, we were able to increase the rating of our chess engine in 557 Elo points (from 1760 to 2317).

The development of multi-objective evolutionary algorithms assisted by surrogate models has increased in the last few years. However, in real-world applications, the high modality and dimensionality that functions have, often causes problems to such models. In fact, if the Pareto optimal set of a multi-objective optimization problem is located in a search space in which the surrogate model is not able to shape the corresponding region, the search could be misinformed and thus converge to wrong regions. Because of this, a considerable amount of research has focused on improving the prediction of the surrogate models by adding the new solutions to the training set and retraining the model. However, when the size of the training set increases, the training complexity can significantly increase. In this paper, we present a surrogate model which maintains the size of the training set, and in which the prediction of the function is improved by using radial basis function networks in a cooperative way. Preliminary results indicate that our proposed approach can produce good quality results when it is restricted to performing only 200, 1,000 and 5,000 fitness function evaluations. Our proposed approach is validated using a set of standard test problems and an airfoil design problem.

The coinciding development of multiobjective evolutionary algorithms (MOEAs) and the emergence of complex problem formulation in the finance and economics areas has led to a mutual interest from both research communities. Since the 1990s, an increasing number of works have thus proposed the application of MOEAs to solve complex financial and economic problems, involving multiple objectives. This paper provides a survey on the state-of-the-art of research, reported in the specialized literature to date, related to this framework. The taxonomy chosen here makes a distinction between the (widely covered) portfolio optimization problem and the other applications in the field. In addition, potential paths for future research within this area are identified.

In recent years, the development of multi-objective evolutionary algorithms (MOEAs) hybridized with mathematical programming techniques has significantly increased. However, most of these hybrid approaches are gradient-based, and tend to require a high number of extra objective function evaluations to estimate the gradient information required. The use of direct search methods—i.e., methods that do not require gradient information—has been, however, less popular in the specialized literature (although such approaches have been used with single-objective evolutionary algorithms). This paper precisely focuses on the design of a hybrid between the wellknownMOEA/ D and Nelder and Mead's algorithm. Clearly, the mathematical programming technique adopted here, acts as a local search mechanism, whose goal is to improve the search performed by MOEA/D. Because of its nature, the proposed local search mechanism can be easily coupled to any other decomposition-based MOEA. Our preliminary results indicate that this sort of hybridization is quite promising for dealing with multi-objective optimization problems (MOPs) having high dimensionality (in decision variable space).

In this paper, we propose a new selection operator (based on a maximin scheme and a clustering technique), which is incorporated into a differential evolution algorithm to solve multi-objective optimization problems. The resulting algorithm is called Maximin-Clustering Differential Evolution (MCDE) and, is validated using standard test problems and performance measures taken from the specialized literature. Our preliminary results indicate that MCDE is able to outperform NSGA-II and that is competitive with a hypervolume-based approach (SMS-EMOA), but at a significantly lower computational cost.

In this paper, we present a surrogate-based multi-objective evolutionary optimization approach to optimize airfoil aerodynamic designs. Our approach makes use of multiple surrogate models which operate in parallel with the aim of combining their features when solving a costly multi-objective optimization problem. The proposed approach is used to solve five multiobjective airfoil aerodynamic optimization problems. We compare the performance of a multi-objective evolutionary algorithm with surrogates with respect to the same approach without using surrogates. Our preliminary results indicate that our proposal can achieve a substantial reduction in the number of objective function evaluations, which has obvious advantages for dealing with expensive objective functions such as those involved in aeronautical optimization problems.

The linear antenna array design problem is one of the most important in electromagnetism. While designing a linear antenna array, the goal of the designer is to achieve the “minimum average side lobe level” and a “null control” in specific directions. In contrast to the existing methods that attempt to minimize a weighted sum of these two objectives considered here, in this paper our contribution is twofold. First, we have considered these as two distinct objectives which are optimized simultaneously in a multi-objective framework. Second, for directivity purposes, we have introduced another objective called the “maximum side lobe level” in the design formulation. The resulting multi-objective optimization problem is solved by using the recently-proposed decomposition-based Multi-Objective Particle Swarm Optimizer (dMOPSO). Our experimental results indicate that the proposed approach is able to obtain results which are better than those obtained by two other state-of-the-art Multi-Objective Evolutionary Algorithms (MOEAs). Additionally, the individual minima reached by dMOPSO outperform those achieved by two single-objective evolutionary algorithms.

In a previous paper presented at CEC'2011, we reported the implementation of a chess engine based on evo- lutionary programming with a selection mechanism that relied on grandmaster's chess games. The objective was to decide the virtual players that would pass to the following generation. Here, we use these same techniques to adjust a larger number of weights (29 in this work instead of the 5 used in the previous one). The aim was to improve the rating of our chess engine. We also introduce here the use of a local search scheme based on the Hooke-Jeeves algorithm, which is adopted to adjust the weights of the best virtual player obtained in the evolutionary process. As our results indicate, this produced a further improvement in the rating of our chess engine. As in our previous work, the material values of the additional pieces considered here are similar to the values known from chess theory.

An emerging trend in the design of multi-objective evolutionary algorithms (MOEAs) is to select individuals through the optimization of a quality assessment indicator. However, the most commonly adopted indicator in current use is the hypervolume which becomes very expensive (computationally speaking) as we increase the number of objectives. In this paper, we propose, instead, the use of another indicator called Δp. Although the Δp indicator is not Pareto compliant, we show here how it can be incorporated into the selection mechanism of an evolutionary algorithm (for that sake, we adopt differential evolution as our search engine) in order to produce a MOEA. The resulting MOEA (called Δp-Differential Evolution, or DDE) is validated using standard test problems and performance indicators reported in the specialized literature. Our results are compared with respect to those obtained by both a Pareto-based MOEA (NSGA-II) and a hypervolume-based MOEA (SMS-EMOA). Our preliminary results indicate that our proposed approach is competitive with respect to these two MOEAs for continuous problems having two and three objective functions. Additionally, our proposed approach is better than NSGA-II and provides competitive results with respect to SMS-EMOA for continuous many-objective problems. However, in this case, the main advantage of our proposal is that its computational cost is significantly lower than that of SMS-EMOA.

Nowadays, the solution of multiobjective optimization problems in aeronautical and aerospace engineering has become a standard practice. These two fields offer highly complex search spaces with different sources of difficulty, which are amenable to the use of alternative search techniques such as metaheuristics, since they require little domain information to operate. From the several metaheuristics available, multiobjective evolutionary algorithms (MOEAs) have become particularly popular, mainly because of their availability, ease of use, and flexibility. This paper presents a taxonomy and a comprehensive review of applications of MOEAs in aeronautical and aerospace design problems. The review includes both the characteristics of the specific MOEA adopted in each case, as well as the features of the problems being solved with them. The advantages and disadvantages of each type of approach are also briefly addressed. We also provide a set of general guidelines for using and designing MOEAs for aeronautical and aerospace engineering problems. In the final part of the paper, we provide some potential paths for future research, which we consider promising within this area.

The Hausdorff distance dHis a widely used tool to measure the distance between different objects in several research fields. Possible reasons for this might be that it is a natural extension of the well-known and intuitive distance between points and/or the fact that dHdefines in certain cases a metric in the mathematical sense. In evolutionary multiobjective optimization (EMO) the task is typically to compute the entire solution set-the so-called Pareto set-respectively its image, the Pareto front. Hence, dHshould, at least at first sight, be a natural choice to measure the performance of the outcome set in particular since it is related to the terms spread and convergence as used in EMO literature. However, so far, dHdoes not find the general approval in the EMO community. The main reason for this is that dHpenalizes single outliers of the candidate set which does not comply with the use of stochastic search algorithms such as evolutionary strategies. In this paper, we define a new performance indicator, Δp, which can be viewed as an “averaged Hausdorff distance” between the outcome set and the Pareto front and which is composed of (slight modifications of) the well-known indicators generational distance (GD) and inverted generational distance (IGD). We will discuss theoretical properties of Δp(as well as for GD and IGD) such as the metric properties and the compliance with state-of-theart multiobjective evolutionary algorithms (MOEAs), and will further on demonstrate by empirical results the potential of Δpas a new performance indicator for the evaluation of MOEAs.

Evolutionary algorithms have been very popular optimization methods for a wide variety of applications. However, in spite of their advantages, their computational cost is still a prohibitive factor in certain real-world applications involving expensive (computationally speaking) fitness function evaluations. In this paper, we depart from the observation that nature's survival of the fittest is not about exact measures of fitness; rather it is about rankings among competing peers. Thus, by exploiting this natural tolerance for imprecision, we propose here a new, fuzzy granules-based approach for reducing the number of necessary function calls involving time consuming real-world problems. Our proposed approach is compared with respect to the standard NSGA-II, using the Set Coverage, Hypervolume and Generational Distance performance measures. Our results indicate that our proposed approach is a very promising alternative for dealing with multi-objective optimization problems involving expensive fitness function evaluations.

Multiobjective optimization problems have been widely addressed using evolutionary computation techniques. However, when dealing with more than three conflicting objectives (the so-called many-objective problems), the performance of such approaches deteriorates. The problem lies in the inability of Pareto dominance to provide an effective discrimination. Alternative ranking methods have been successfully used to cope with this issue. Nevertheless, the high selection pressure associated with these approaches usually leads to diversity loss. In this study, we focus on parallel genetic algorithms, where multiple partially isolated subpopulations are evolved concurrently. As in nature, isolation leads to speciation, the process by which new species arise. Thus, evolving multiple subpopulations can be seen as a potential source of diversity and it is known to improve the search performance of genetic algorithms. Our experimental results suggest that such a behavior, integrated with an effective ranking, constitutes a suitable approach for many objective optimization.

In this paper, we assess the convenience of applying a previously proposed interactive method to solve three aero dynamic airfoil shape optimization problems with 2, 3, and 6 objectives, respectively. The expensive simulations required to evaluate the objective functions makes these problems an excellent example in which the use of interactive methods is very advantageous. First, the search can be focused on the decision maker's region of interest, saving this way, valuable function evaluations. Second, the preference relation used in the interactive method helps to deal with a large number of objectives since it is able to rank incomparable nondominated solutions. The experimental evaluation reveals that in the three problems studied, the interactive method achieved a better final solution than a traditional a posteriori method with no preferences. Nevertheless, in the problem with 6 objectives, only 3 of them were improved. A possible explanation for this is that local optima become harder to overcome when the size of the region of interest is very small. Additional experiments confirmed that the convergence is deteriorated if very small regions of interest are used.

This paper presents a Multi-Region Differential Evolution (MRDE) algorithm as an extension of a classical version of differential evolution (DE) (i.e., as an extension of DE/rand-to-best/1/exp). MRDE is designed to simultaneously search on different and evenly distributed sub-regions on the whole search space. The number and extent of the search regions change during the execution of the algorithm, in such a way that, at the final stage of the evolutionary process, only one region remains (i.e., the whole search space). Our proposed MRDE is compared with respect to the classical DE algorithm on a set of well-known benchmark problems. The results achieved show enough evidence of the benefits of distributing the population of vectors when dealing with large-scale optimization problems.

This paper proposes an algorithm for dealing with nonlinear and unconstrained multi-objective optimization problems (MOPs). The proposed algorithm adopts a nonlinear simplex search scheme in order to obtain multiple approximations of the Pareto optimal set. The search is directed by a well-distributed set of weighted vectors. Each weighted vector defines a scalarization problem which is solved by deforming a simplex according to the movements described by Nelder and Mead's method. The simplex is constructed with a set of solutions which minimize different scalarization problems defined by a set of neighbor weighted vectors. The solutions found in the search are used to update a set of solutions considered to be the minima for each separate problem. In this way, the proposed algorithm collectively obtains multiple trade-offs among the different conflicting objectives, while maintaining a well distributed set of solutions along the Pareto front. The main aim of this work is to show that a well-designed strategy using just mathematical programming techniques can be competitive with respect to a state-of-the-art multi-objective evolutionary algorithm.

This paper proposes a method for tuning the weights of the evaluation function of a chess program whose search engine is based on evolutionary programming. In our proposed approach, each individual in the population of the evolutionary algorithm represents a virtual player with specific weights of its evaluation function. This differs from most of the previous approaches reported in the literature, in which normally a tournament between virtual players is held, and the final result (win, loss or draw) is used to decide which players will pass to the following generation. The selection mechanism of our proposed algorithm uses games from chess grandmasters to decide which virtual player will pass to the following generation. Our results indicate that the weight values obtained by our approach are similar to the values known from chess theory. Additionally, the standard deviation from the different runs performed, are lower than those reported by authors of previous related approaches.

The simplicity and success of particle swarm optimization (PSO) algorithms, has motivated researchers to extend the use of these techniques to the multi-objective optimization field. This paper presents a multi-objective particle swarm optimization (MOPSO) algorithm based on a decomposition approach, which is intended for solving continuous and unconstrained multi-objective optimization problems (MOPs). The proposed decomposition-based multi-objective particle swarm optimizer (dMOPSO), updates the position of each particle using a set of solutions considered as the global best according to the decomposition approach. dMOPSO is mainly characterized by the use of a memory reinitialization process which aims to provide diversity to the swarm. Our proposed approach is compared with respect to two decomposition-based multi-objective evolutionary algorithms (MOEAs) which are representative of the state-of-the-art in the area. Our results indicate that our proposed approach is competitive and it outperforms the two MOEAs with respect to which it was compared in most of the test problems adopted.

The six papers in this special issue are representative of the current research trends in differential evolution.

In this paper, we study the influence of the number of objectives of a continuous multiobjective optimization problem on its hardness for evolution strategies which is of particular interest for many-objective optimization problems. To be more precise, we measure the hardness in terms of the evolution (or convergence) of the population toward the set of interest, the Pareto set. Previous related studies consider mainly the number of nondominated individuals within a population which greatly improved the understanding of the problem and has led to possible remedies. However, in certain cases this ansatz is not sophisticated enough to understand all phenomena, and can even be misleading. In this paper, we suggest alternatively to consider the probability to improve the situation of the population which can, to a certain extent, be measured by the sizes of the descent cones. As an example, we make some qualitative considerations on a general class of uni-modal test problems and conjecture that these problems get harder by adding an objective, but that this difference is practically not significant, and we support this by some empirical studies. Further, we address the scalability in the number of objectives observed in the literature. That is, we try to extract the challenges for the treatment of many-objective problems for evolution strategies based on our observations and use them to explain recent advances in this field.

In this paper, two novel evolutionary approaches for many-objective optimization are proposed. These algorithms integrate a fine-grained ranking of solutions to favor convergence, with explicit methodologies for diversity promotion in order to guide the search towards a representative approximation of the Pareto-optimal surface. In order to validate the proposed algorithms, we performed a comparative study where four state-of-the-art representative approaches were considered. In such a study, four well-known scalable test problems were adopted as well as six different problem sizes, ranging from 5 to 50 objectives. Our results indicate that our two proposed algorithms consistently provide good convergence as the number of objectives increases, outperforming the other approaches with respect to which they were compared.

The performance of ant colony optimization (ACO) algorithms significantly improves when hybridized with local search procedures which strongly bias the search towards promising regions of the search space. In this work, we study a recently proposed Memory based ACO algorithm (MACO) which incorporates some tabu search principles into the solution construction process. This algorithm has also been hybridized with two local search procedures: 2-opt (M-ACO-2opt) and Tabu Search (M-ACO-TS). The performances of the two hybrid versions of M-ACO are analyzed on a set of instances of the Quadratic Assignment Problem (QAP). The results show that the hybrid versions of M-ACO are able to improve the quality of the best known solutions for several of the instances studied.

In this work we present a simple way to introduce gradient-based information as a means to improve the search performed by a multi-objective evolutionary algorithm (MOEA). Our proposal can be easily incorporated into any MOEA, and is able to improve its performance when solving continuous bi-objective problems. We propose a novel mechanism to control the balance between the local search, and the global search performed by a MOEA. We discuss the advantages of the proposed method and its possible use when dealing with more objectives. Finally, we provide some guidelines regarding the use of our proposed approach.

Diversity plays an important role in evolutionary multi-objective optimization. Because of this, a number of density estimators (i.e., mechanisms that help to maintain diversity) have been proposed since the early days of multi-objective evolutionary algorithms (MOEAs). Fitness sharing and niching were among the most popular density estimator used with non-elitist MOEAs, but their main drawback was their high dependence on the niche radius, which was normally difficult to set. In recent years, the use of external archives to store the nondominated solutions found by an elitist MOEA has become popular. This has motivated an important amount of research related to archiving techniques for MOEAs. In this paper, we contribute to such literature by introducing a new archiving strategy based on the Convex Hull of Individual Minima (CHIM). Our proposed approach is compared with respect to two competitive MOEAs (NSGA-II and SPEA2) using standard test problems and performance measures taken from the specialized literature.

In this paper, we present a novel Multi-Objective Evolutionary Algorithm (MOEA) called MODE-LD+SS, which combines Differential Evolution with local dominance and a scalar selection mechanism for improving both its convergence rate and its distribution of solutions along the Pareto front. In order to assess the performance of the proposed approach, we use a set of standard test functions and performance measures taken from the specialized literature. Results are compared with respect to three MOEAs representative of the state-of-the-art in the area: NSGA-II, SPEA2, and MOEA/D.

In many applications it can be advantageous for the decision maker to have multiple options available for a possible realization of the project. One way to increase the number of interesting choices is in certain cases to consider in addition to the optimal solution x∗also nearly optimal or approximate solutions which differ in the design space from x∗by a certain value. In this paper we address the efficient computation and discretization of the set E of ∊-approximate solutions for scalar optimization problems. For this we will suggest two strategies to archive and update the data coming from the generation process of the search procedure, and will use Differential Evolution coupled with the new archivers for the computation of E. Finally, we will demonstrate the behavior of the archiver empirically on some academic functions as well as on two models related to space mission design.

In this paper, we present an approach in which a local search mechanism is coupled to a multi-objective evolutionary algorithm. The local search mechanism is assisted by a meta-model based on support vector machines. Such a mechanism consists of two phases: the first one involves the use of an aggregating function which is defined by different weighted vectors. For the (scalar) optimization task involved, we adopt a non-gradient mathematical programming technique: the Hooke-Jeeves method. The second phase computes new solutions departing from those obtained in the first phase. The local search engine generates a set of solutions which are used in the evolutionary process of our algorithm. The preliminary results indicate that our proposed approach is quite promising.

Recently, a convergence proof of stochastic search algorithms toward finite size Pareto set approximations of continuous multi-objective optimization problems has been given. The focus was on obtaining a finite approximation that captures the entire solution set in some suitable sense, which was defined by the concept of epsilon-dominance. Though bounds on the quality of the limit approximation-which are entirely determined by the archiving strategy and the value of epsilon-have been obtained, the strategies do not guarantee to obtain a gap free approximation of the Pareto front. That is, such approximations A can reveal gaps in the sense that points f in the Pareto front can exist such that the distance of f to any image point F(a), a epsilon A, is "large." Since such gap free approximations are desirable in certain applications, and the related archiving strategies can be advantageous when memetic strategies are included in the search process, we are aiming in this work for such methods. We present two novel strategies that accomplish this task in the probabilistic sense and under mild assumptions on the stochastic search algorithm. In addition to the convergence proofs, we give some numerical results to visualize the behavior of the different archiving strategies. Finally, we demonstrate the potential for a possible hybridization of a given stochastic search algorithm with a particular local search strategy-multi-objective continuation methods-by showing that the concept of epsilon-dominance can be integrated into this approach in a suitable way.

To evaluate the search capabilities of a multiobjective algorithm, the usual approach is to choose a benchmark of known problems, to perform a fixed number of function evaluations, and to apply a set of quality indicators. However, while real problems could have hundreds or even thousands of decision variables, current benchmarks are normally adopted with relatively few decision variables (normally from 10 to 30). Furthermore, performing a constant number of evaluations does not provide information about the effort required by an algorithm to get a satisfactory set of solutions; this information would also be of interest in real scenarios, where evaluating the functions defining the problem can be computationally expensive. In this paper, we study the effect of parameter scalability in a number of state-of-the-art multiobjective metaheuristics. We adopt a benchmark of parameter-wise scalable problems (the Zitzler-Deb-Thiele test suite) and analyze the behavior of eight multiobjective metaheuristics on these test problems when using a number of decision variables that range from 8 up to 2048. By using the hypervolume indicator as a stopping condition, we also analyze the computational effort required by each algorithm in order to reach the Pareto front. We conclude that the two analyzed algorithms based on particle swarm optimization and differential evolution yield the best overall results.

In this paper, we propose and investigate a new local search strategy for multiobjective memetic algorithms. More precisely, we suggest a novel iterative search procedure, known as theHill Climber with Sidestep(HCS), which is designed for the treatment of multiobjective optimization problems, and show further two possible ways to integrate the HCS into a given evolutionary strategy leading to new memetic (or hybrid) algorithms. The pecularity of the HCS is that it is intended to be capable both moving toward and along the (local) Pareto set depending on the distance of the current iterate toward this set. The local search procedure utilizes the geometry of the directional cones of such optimization problems and works with or without gradient information. Finally, we present some numerical results on some well-known benchmark problems, indicating the strength of the local search strategy as a standalone algorithm as well as its benefit when used within a MOEA. For the latter we use the state of the art algorithms Nondominated Sorting Genetic Algorithm-II and Strength Pareto Evolutionary Algorithm 2 as base MOEAs.

This work introduces a hybrid between an elitist multi-objective evolutionary algorithm and a gradient-based descent method, which is applied only to certain (selected) solutions. Our proposed approach requires a low number of objective function evaluations to converge to a few points in the Pareto front. Then, the rest of the Pareto front is reconstructed using a method based on rough sets theory, which also requires a low number of objective function evaluations. Emphasis is placed on the effectiveness of our proposed hybrid approach when increasing the number of decision variables, and a study of the scalability of our approach is also presented.

In this paper, we propose a new selection criterion for candidate solutions to a constrained optimization problem. Such a selection mechanism is incorporated into a differential evolution (DE) algorithm. This DE approach is then hybridized with an operator based on the Nelder-Mead method, whose aim is to speed up convergence towards good solutions. The proposed approach is called “Hybrid of Differential Evolution and the Simplex Method for Constrained Optimization Problems” (HDESMCO), and is validated using a well-know benchmark for constrained evolutionary optimization. The results indicate that our proposed approach produces solutions whose quality is competitive with respect to those generated by three evolutionary algorithms from the state-of-the-art (improved stochastic ranking, diversity-DE and Generalized Differential Evolution), but requiring a lower number of objective function evaluations.

Since the introduction of the particle swarm optimization (PSO) algorithm, a considerable amount of research has been devoted to devise mechanisms that can control its possible premature convergence. The most common approach to deal with premature convergence in PSO consists of controlling (e.g., by limiting) the velocity of a particle. In this paper, we present a method that consists of limiting the velocity of a particle using the elements of a sequence of a geometric series. This approach is not only simplest than the current available methods, but also presents competitive results, and even better convergence in some cases, than two other PSO-based approaches. Additionally, the proposed approach provides more flexibility to balance between exploration or exploitation, through the tuning of a single parameter.

This paper presents a quantitative analysis of different preference relations proposed to deal with problems with a high number of objectives. Since the relations stress different subsets of the Pareto front, we based the comparison on the Tchebycheff distance of the approximation set to the "knee" of the Pareto front. Additionally, the convergence induced by the preference relations is studied by analyzing the generational distance observed at each generation of the search. The results show that some preference relations contribute to converge quickly to the Pareto front, but they promote the generation of solutions far from the knee region. Moreover, even if a preference relation generates solutions near the knee, there exists a trade-off between convergence and the extension of

In this paper we develop evolutionary strategies for numerical continuation which we apply to scalar and multi-objective optimization problems. To be more precise, we will propose two different methods-an embedding algorithm and a multi-objectivization approach-which are designed to follow an implicitly defined curve where the aim can be to detect the endpoint of the curve (e.g., a root finding problem) or to approximate the entire curve (e.g., the Pareto set of a multi-objective optimization problem). We demonstrate that the novel approaches are very robust in finding the set of interest (point or curve) on several examples.

This study addresses the solution of jobshop scheduling problems using differential evolution (DE). The issue of representing permutations through real numbers constitutes the key issue for developing an efficient implementation. Several techniques are empirically validated on problem instances traditionally adopted in the specialized literature. We also present a simple hybridization of DE with tabu search, which produces significant performance gains.

This paper presents a novel boundary approach that is included as a constraint-handling technique in an algorithm inspired by the ant colony metaphor. The necessity of approaching the boundary between the feasible and infeasible search space for many constrained optimization problems is a paramount challenge for every constraint-handling technique. Our proposed technique precisely focuses the search on the boundary region and can be either used alone or in combination with other constraint-handling techniques depending on the type and number of problem constraints. For validation purposes, an algorithm inspired by the ant colony metaphor is adopted as our search engine that works following one of the principles of the ant colony approach, i.e., a population of agents iteratively, cooperatively, and independently search for a solution. Each ant in the distributed algorithm applies a simple mutation-like operator, which explores the neighborhood region of a particular point in the search space (individual search level). The operator is designed for exploring the boundary between the feasible and infeasible search space. In addition, each ant obtains global information from the colony in order to exploit the most promising regions of the search space (cooperation level). We compare our proposed approach with respect to a well-known constraint-handling technique that is representative of the state-of-the-art in the area, using a set of standard test functions.

Much of the computational complexity in employing evolutionary algorithms as optimization tool is due to the fitness function evaluation that may either not exist or be computationally very expensive. With the proposed approach, the expensive fitness evaluation step is replaced by an approximate model. An intelligent guided technique via an adaptive fuzzy similarity analysis for fitness granulation is used to decide on use of expensive function evaluation and dynamically adapt the predicted model. In order to avoid tuning parameters in this approach, a fuzzy supervisor as auto-tuning algorithm is employed with three inputs. The proposed method is then applied to three traditional optimization benchmarks with four different choices for the dimensionality of the search apace. Effect of number of granules on rate of convergence is also studied. In comparison with standard application of evolutionary algorithms, statistical analysis confirms that the proposed approach demonstrates an ability to reduce the computational complexity of the design problem without sacrificing performance. Furthermore, the auto-tuning of the fuzzy supervisory removes the need for exact parameter determination.

Some real-world optimization problems have hundreds or even thousands of decision variables. However, the effect that the scalability of parameters has in modern multi-objective metaheuristic algorithms has not been properly studied (the current benchmarks are normally adopted with ten to thirty decision variables). In this paper, we adopt a benchmark of parameter-wise scalable problems (the ZDT test problems) and analyze the behavior of six multi-objective metaheuristics on these test problems when using a number of decision variables that goes from 8 up to 2048. The computational effort required by each algorithm in order to reach the true Pareto front is also analyzed. Our study concludes that a particle swarm algorithm provides the best overall performance, although it has difficulties in multifrontal problems.

In the field of single-objective optimization, hybrid variants of gradient-based methods and evolutionary algorithms have been shown to perform better than an evolutionary method by itself. This same idea has been recently used in Evolutionary Multiobjective Optimization (EMO), obtaining also very promising results. In most cases, gradient information is used along the whole process, which involves a high computational cost, mainly related to the computation of the step lengths required. In contrast, in this paper we propose the use of gradient information only at the beginning of the search process. We will show that this sort of scheme maintains results of good quality while considerably decreasing the computational cost. In our work, we adopt a steepest descent method to generate some nondominated points which are then used to seed the initial population of a multi-objective evolutionary algorithm (MOEA), which will spread them along the Pareto front. The MOEA adopted in our case is the NSGA-II, which is representative of the state-of-the-art in the area. To validate our proposal, we adopt box-constrained continuous problems (the ZDT test suite). The gradients required are approximated using quadratic regressions. Our proposed approach performs a total of 2000 objective function evaluations, which is much lower than the number of evaluations normally adopted with the ZDT test suite in the specialized literature. Our results are compared with respect to the “pure” NSGA-II (i.e., without using gradient-based information) so that the potential benefit of these initial solutions fed into the population can be properly assessed.

In this paper a new approach to solve constrained multi-objective problems by way of evolutionary multi-objective optimization is introduced. In contrast to former evolutionary approaches, which amalgamate objective space dominance relations with feasibility of solutions considered in the design spaces, the hereby suggested approach relies solely on objective space based analysis. It is shown in this paper that considering the violation of constraints within the design space is problematic as it may lead to misleading conclusions. Moreover, the current approach is inherently capable of dealing with constraints that are imposed directly in the objective space.

This paper introduces two new algorithms to reduce the number of objectives in a multiobjective problem by identifying the most conflicting objectives. The proposed algorithms are based on a feature selection technique proposed by Mitra et. al. [11]. One algorithm is intended to determine the minimum subset of objectives that yields the minimum error possible, while the other finds a subset of objectives of a given size that yields the minimum error. To validate these algorithms we compare their results against those obtained by two similar algorithms recently proposed. The comparative study shows that our algorithms are very competitive with respect to the reference algorithms. Additionally, our approaches require a lower computational time. Also, in this study we propose to use the inverted generational distance to evaluate the quality of a subset of objectives.

In recent years, the development of multi-objective evolutionary algorithms (MOEAs) hybridized with mathematical programming techniques has significantly increased. However, most of these hybrid approaches are gradient-based, and tend to require a high number of extra objective function evaluations to estimate the gradient information required. The use of nonlinear optimization approaches taken from the mathematical programming literature has been, however, less popular (although such approaches have been used with single-objective evolutionary algorithms). This paper precisely focuses on the design of a hybrid between a well-known MOEA (the NSGA-II) and two direct search methods taken from the mathematical programming literature (Nelder and Mead.s method and the golden section algorithm). The idea is to combine the explorative power of the evolutionary algorithm with the exploitative power of the direct search methods previously indicated (one is used for unidimensional functions and the other for multidimensional functions). Clearly, these mathematical programming techniques act as local search engines, whose goal is to refine the search performed by the MOEA. Our preliminary results indicate that this sort of hybridization is quite promising.

This paper presents an approach in which a multi-objective evolutionary algorithm (MOEA) is coupled to a surrogate method in order to explore the search space in an efficient manner. A small comparative study among three surrogate methods is conducted: an artificial neural network (ANN), a radial basis function (RBF) and a support vector machine (SVM). The winner in this comparative study was the SVM. However, our results indicated that the spread of solutions achieved by our surrogate-based MOEA was poor. Thus, we decided to introduce a second phase to the algorithm in which it is hybridized with the rough sets in order to improve the spread of solutions and help to reach the true Pareto front. We show that our proposed hybrid approach only requires 2,000 fitness function evaluations in order to solve test problems with up to 30 decision variables.

In this work we study the convergence of generic stochastic search algorithms toward the entire set of approximate solutions of continuous multi-objective optimization problems. Since the dimension of the set of interest is typically equal to the dimension of the parameter space, we focus on obtaining a finite and tight approximation, measured by the Hausdorff distance. Under mild assumptions about the process to generate new candidate solutions, the limit approximation set will be determined entirely by the archiving strategy. We propose and investigate a novel archiving strategy theoretically and empirically. For this, we analyze the convergence behavior of the algorithm, yielding bounds on the obtained approximation quality as well as on the cardinality of the resulting approximation, and present some numerical results.

In this paper we propose a novel iterative search procedure for multi-objective optimization problems. The iteration process -- though derivative free -- utilizes the geometry of the directional cones of such optimization problems, and is capable both of moving toward and along the (local) Pareto set depending on the distance of the current iterate toward this set. Next, we give one possible way of integrating this local search procedure into a given EMO algorithm resulting in a novel memetic strategy. Finally, we present some numerical results on some well-known benchmark problems indicating the strength of both the local search strategy as well as the new hybrid approach.

This paper presents a new algorithm that approximates real function evaluations using supervised learning with a surrogate method called support vector machine (SVM). We perform a comparative study among different leader selection schemes in a Multi-Objective Particle Swarm Optimizer (MOPSO), in order to determine the most appropriate approach to be adopted for solving the sort of problems of our interest. The resulting hybrid presents a poor spread of solutions, which motivates the introduction of a second phase to our algorithm, in which an approach called rough sets is adopted in order to improve the spread of solutions along the Pareto front. Rough sets are used as a local search engine, which is able to generate solutions in the neighborhood of the nondominated solutions previously generated by the surrogate-based algorithm. The resulting approach is able to generate reasonably good approximations of the Pareto front of problems of up to 30 decision variables with only 2,000 fitness function evaluations. Our results are compared with respect to the NSGA-II, which is a multi-objective evolutionary algorithm representative of the state-of-the-art in the area.

This paper deals with the optimal computation of finite field exponentiation, which is a well-studied problem with many important applications in the areas of error-correcting codes and cryptography. It has been shown that the optimal computation of finite field exponentiation is a problem which is closely related to finding a suitable addition chain with the shortest possible length. However, it is also known that obtaining the shortest addition chain for a given arbitrary exponent is an NP-hard problem. As a consequence, heuristics are an obvious choice to compute field exponentiation with a semi-optimal number of underlying arithmetic operations. In this paper, we propose the use of an artificial immune system to tackle this problem. Particularly, we study the problem of finding both the shortest addition chains for exponentsewith moderate size (i.e., with a length of less than 20 bits), and for the huge exponents typically adopted in cryptographic applications, (i.e., in the range from 128 to 2048 bits).

This article introduces the Non-Parametric Real-valued Estimation Distribution Algorithm (NOPREDA), and its application to constrained optimization problems. NOPREDA approximates the target probability density function by building the cumulative empirical distribution of the decision variables. Relationships and structure among the data is modeled through a rank correlation matrix (Spearmans statistics). The procedure to induce a target rank correlation matrix into the new population is described. NOPREDA is used to solve constrained optimization problems. Three constraint handling techniques are investigated: truncation selection, feasibility tournament, and Stochastic Ranking. NOPREDA’s performance is competitive in problems with inequality constraints. However, a mechanism for properly handling equality constraints remains as part of our future research work.

This paper presents an enhanced Particle Swarm Optimizer approach, which is designed to solve numerical constrained optimization problems. The approach uses a single method to handle different types of constraints (linear, nonlinear, equality or inequality) and it incorporates a shakemechanism and a dual population in an attempt to overcome the problem of premature convergence to local optima. The proposed algorithm is validated using standard test functions taken from the specialized literature and is compared with respect to algorithms representative of the state-of-the-art in the area. Our preliminary results indicate that our proposed approach is a highly competitive alternative to solve constrained optimization problems.

In this paper we present a boundary search based ACO algorithm for solving nonlinear constrained optimization problems. The aim of this work is twofold. Firstly, we present a modified search engine which implements a boundary search approach based on a recently proposed ACO metaheuristic for continuos problems. Secondly, we propose the incorporation of the stochastic ranking technique to deal with feasible and infeasible solutions during the search which focuses on the boundary region. In our experimental study we compare the overall performance of the proposed ACO algorithm by including two different complementary constraint-handling techniques: a penalty function and stochastic ranking. In addition, we include in our comparison of results the Stochastic Ranking algorithm, which was originally implemented using an Evolution Strategy as its search engine.

This paper provides a state-of-the-art survey of applications of multi-objective evolutionary algorithms in economics and finance reported in the specialized literature. A taxonomy of applications within this area is proposed, and a brief review of the most representative research reported to date is then provided. In the final part of the paper, some potential paths for future research within this area are identified.

In this paper, we present an ant system algorithm variant designed to solve the job shop scheduling problem. The proposed approach is based on a recent biological study which showed that natural ants can count their steps when they build the path between the nest and their food source. Experiments using a set of well-known job shop scheduling problems and a comparison against state-of-the-art techniques show that the proposed approach can reduce the number of evaluations performed without a degradation of performance. Additionally, our proposed approach reduces the number of parameters that need to be tuned by the user (specifically the parameters that balance the importance between the pheromone trail and heuristic values), with respect to the original ant system algorithm.

In this paper, we propose the combination of different optimization techniques in order to solve "hard" two- and three-objective optimization problems at a relatively low computational cost. First, we use the ε-constraint method in order to obtain a few points over (or very near of) the true Pareto front, and then we use an approach based on rough sets to spread these solutions, so that the entire Pareto front can be covered. The constrained single-objective optimizer required by the ε-constraint method, is the cultured differential evolution, which is an efficient approach for approximating the global optimum of a problem with a low number of fitness function evaluations. The proposed approach is validated using several difficult multi-objective test problems, and our results are compared with respect to a multi-objective evolutionary algorithm representative of the state-of-the-art in the area: the NSGA-II.

Recently, a convergence proof of stochastic search algorithms toward finite size Pareto set approximations of continuous multi-objective optimization problems has been given. The focus was on obtaining a finite approximation that captures the entire solution set in some suitable sense, which was defined by the concept of ε-dominance. Though bounds on the quality of the limit approximation -- which are entirely determined by the archiving strategy and the value of ε -- have been obtained, the strategies do not guarantee to obtain a gap-free Pareto front approximation. Since such approximations are desirable in certain applications, and the related archiving strategies can be advantageous when memetic strategies are included into the search process, we are aiming in this work for such methods. We present two novel strategies that accomplish this task in the probabilistic sense and under mild assumptions on the stochastic search algorithm. In addition to the convergence proofs we give somenumerical results to visualize the behavior of the different archiving strategies.

In this paper we introduce line search strategies originating from continuous optimization for the realization of the guidance mechanism in particle swarm optimization for scalar optimization problems. Since these techniques are well-suited for-but not restricted to-local search the resulting algorithm can be considered to be memetic. Further, we will use the same techniques for the construction of a new variant of a hill climber. We will discuss possible realizations and will finally present some numerical results indicating the strength of the two algorithms

Abstract In this paper, we introduce MRMOGA (Multiple Resolution Multi‐Objective Genetic Algorithm), a new parallel multi‐objective evolutionary algorithm which is based on an injection island approach. This approach is characterized by adopting an encoding of solutions which uses a different resolution for each island. This approach allows us to divide the decision variable space into well‐defined overlapped regions to achieve an efficient use of multiple processors. Also, this approach guarantees that the processors only generate solutions within their assigned region. In order to assess the performance of our proposed approach, we compare it to a parallel version of an algorithm that is representative of the state‐of‐the‐art in the area, using standard test functions and performance measures reported in the specialized literature. Our results indicate that our proposed approach is a viable alternative to solve multi‐objective optimization problems in parallel, particularly when dealing with large search spaces. Copyright © 2006 John Wiley & Sons, Ltd.

Efficiency has become one of the main concerns in evolutionary multiobjective optimization during recent years. One of the possible alternatives to achieve a faster convergence is to use a relaxed form of Pareto dominance that allows us to regulate the granularity of the approximation of the Pareto front that we wish to achieve. One such relaxed forms of Pareto dominance that has become popular in the last few years is epsilon-dominance, which has been mainly used as an archiving strategy in some multiobjective evolutionary algorithms. Despite its advantages, epsilon-dominance has some limitations. In this paper, we propose a mechanism that can be seen as a variant of epsilon-dominance, which we call Pareto-adaptive epsilon-dominance (paepsilon-dominance). Our proposed approach tries to overcome the main limitation of epsilon-dominance: the loss of several nondominated solutions from the hypergrid adopted in the archive because of the way in which solutions are selected within each box.

In this paper, we present a Differential-Evolution based approach to solve constrained optimization problems. The aim of the approach is to increase the probability of each parent to generate a better offspring. This is done by allowing each solution to generate more than one offspring but using a different mutation operator which combines information of the best solution in the population and also information of the current parent to find new search directions. Three selection criteria based on feasibility are used to deal with the constraints of the problem and also a diversity mechanism is added to maintain infeasible solutions located in promising areas of the search space. The approach is tested in a set of test problems proposed for the special session on Constrained Real Parameter Optimization. The results obtained are discussed and some conclusions are established.

This paper presents a new multi-objective evolutionary algorithm (MOEA) based on differential evolution and rough sets theory. The proposed approach adopts an external archive in order to retain the nondominated solutions found during the evolutionary process. Additionally, the approach also incorporates the concept of paε-dominance to get a good distribution of the solutions retained. The main idea of the approach is to use differential evolution (DE) as our main search engine, trying to translate its good convergence properties exhibited in single-objective optimization to the multi-objective case. Rough sets theory is adopted in a second stage of the search in order to improve the spread of the nondominated solutions that have been found so far. Our hybrid approach is validated using standard test functions and metrics commonly adopted in the specialized literature. Our results are compared with respect to the NSGA-II, which is a MOEA representative of the state-of-the-art in the area.

In this paper, we present an empirical comparison of some Differential Evolution variants to solve global optimization problems. The aim is to identify which one of them is more suitable to solve an optimization problem, depending on the problem's features and also to identify the variant with the best performance, regardless of the features of the problem to be solved. Eight variants were implemented and tested on 13 benchmark problems taken from the specialized literature. These variants vary in the type of recombination operator used and also in the way in which the mutation is computed. A set of statistical tests were performed in order to obtain more confidence on the validity of the results and to reinforce our discussion. The main aim is that this study can help both researchers and practitioners interested in using differential evolution as a global optimizer, since we expect that our conclusions can provide some insights regarding the advantages or limitations of each of the variants studied.

In this paper, we propose a dynamic mechanism to vary the probability by which fitness inheritance is applied throughout the run of a multi-objective particle swarm optimizer, in order to obtain a greater reduction in computational cost (than the obtained with a fixed probability), without dramatically affecting the quality of the results. The results obtained show that it is possible to reduce the computational cost by 32% without affecting the quality of the obtained Pareto front.

Whereas multiobjective evolutionary algorithms have reached certain effectiveness in solving many real-world problems efficiency still remains as an open problem. One choice to reduce the execution time of the multiobjective evolutionary algorithms is their parallelization. This paper introduces a parallel MOEA which is based on the island model with heterogeneous nodes. This algorithm is characterized by encoding the solutions using a different resolution for each island. In this way, the search space is divided into well-defined overlapped regions in decision variable space.

In this paper, we present an empirical study whose aim is twofold: (1) to analyze the on-line behavior of two state-of-the-art approaches for constrained optimization, whose results provided in a well-known benchmark were competitive, in order to identify features of a problem which makes it difficult to solve when using an evolutionary algorithm and (2) to propose a new set of problems whose features cover those sources of difficulty. The on-line behavior analyzed consists on using three performance measures to know how fast the technique reaches the feasible region and to also know the capabilities of the algorithm to improve feasible solutions previously found. Besides, we analyze the ability of the approaches to maintain diversity (to have solutions inside and outside the feasible region as well). Based on the obtained results we propose a set of eleven test problems (either artificial or real-world problems) taken from the literature in order to re-test the approaches. The results are discussed and some conclusions are drawn.

In this paper, we study the use of fitness inheritance and approximation techniques to reduce the number of fitness evaluations into a PSO-based multi-objective algorithm previously proposed by the authors. Fifteen fitness inheritance techniques and four approximation techniques are applied to a set of four well-known test functions taken from the multi-objective optimization literature. A comparison of the best techniques found against other PSO-based multi-objective approaches is carried out using other test functions. The obtained results show a good performance of the enhancement techniques proposed.

In this paper we propose a cultural algorithm, where different knowledge sources modify the variation operator of a differential evolution algorithm. Differential evolution is used as a basis for the population, variation and selection processes. The experiments performed show that the cultured differential evolution is able to reduce the number of fitness function evaluations needed to obtain a good aproximation of the optimum value in constrained real-parameter optimization. Comparisons are provided with respect to three techniques that are representative of the state-of-the-art in the area.

In this paper, we incorporate a diversity mechanism to the differential evolution algorithm to solve constrained optimization problems without using a penalty function. The aim is twofold: (1) to allow infeasible solutions with a promising value of the objective function to remain in the population and also (2) to increase the probabilities of an individual to generate a better offspring while promoting collaboration of all the population to generate better solutions. These goals are achieved by allowing each parent to generate more than one offspring. The best offspring is selected using a comparison mechanism based on feasibility and this child is compared against its parent. To maintain diversity, the proposed approach uses a mechanism successfully adopted with other evolutionary algorithms where, based on a parameter Sr a solution (between the best offspring and the current parent) with a better value of the objective function can remain in the population, regardless of its feasibility. The proposed approach is validated using test functions from a well-known benchmark commonly adopted to validate constraint-handling techniques used with evolutionary algorithms. The statistical results obtained by the proposed approach are highly competitive (based on quality, robustness and number of evaluations of the objective function) with respect to other constraint-handling techniques, either based on differential evolution or on other evolutionary algorithms, that are representative of the state-of-the-art in the area. Finally, a small set of experiments were made to detect sensitivity of the approach to its parameters.

Summary form only given. There has been an increasing interest in using heuristic search algorithms based on natural selection (the so called "evolutionary algorithms") for solving a wide variety of problems. As in any other discipline, research on evolutionary algorithms has become more specialized over the years, giving rise to a number of subdisciplines. This paper deals with one of the emerging subdisciplines that have become very popular due to its wide applicability: evolutionary multi-objective optimization (EMO). EMO refers to the use of evolutionary algorithms (or even other biologically inspired heuristics) to solve problems with two or more (often conflicting) objectives. Unlike traditional (single objective) problems, multi-objective optimization problems normally have more than one possible solution. Thus, traditional evolutionary algorithms (e.g., genetic algorithms) need to be modified in order to deal with such problems. This talk provides a general overview of this field, including its historical origins, its most significant developments, some of its most important application areas and its current challenges.

In this paper, we propose to incorporate the concept of fitness inheritance into a multi-objective particle swarm optimizer previously proposed by us, in order to reduce the number of function evaluations performed. Four well-known test functions taken from the multi-objective optimization literature are used to evaluate the performance of the proposed approach. The results indicate a very good performance of the fitness inheritance technique, mainly when it is applied with a low probability, in which case the quality of the obtained results is even improved.

In this paper, we propose a new mechanism to maintain diversity in multi-objective optimization problems. The proposed mechanism is based on the use of stripes that are applied on objective function space and that is independent of the search engine adopted to solve the multi-objective optimization problem. In order to validate the proposed approach, we included it in a multi-objective particle swarm optimizer. Our approach was compared with respect to two multi-objective evolutionary algorithms, which are representative of the state-of-the-art in the area. The results obtained indicate that our proposed mechanism is a viable alternative to maintain diversity in the context of multi-objective optimization.

This work presents a simple multimembered evolution strategy to solve global nonlinear optimization problems. The approach does not require the use of a penalty function. Instead, it uses a simple diversity mechanism based on allowing infeasible solutions to remain in the population. This technique helps the algorithm to find the global optimum despite reaching reasonably fast the feasible region of the search space. A simple feasibility-based comparison mechanism is used to guide the process toward the feasible region of the search space. Also, the initial stepsize of the evolution strategy is reduced in order to perform a finer search and a combined (discrete/intermediate) panmictic recombination technique improves its exploitation capabilities. The approach was tested with a well-known benchmark. The results obtained are very competitive when comparing the proposed approach against other state-of-the art techniques and its computational cost (measured by the number of fitness function evaluations) is lower than the cost required by the other techniques compared.

Entropy-based measures, such as mutual information and normalized mutual information are investigated as tools for similarity measures between the target and evolving circuit. Three fitness functions are built over a primitive one. We show that the search landscape of normalized mutual information looks more amenable for evolutionary computation algorithms than simple mutual information. The evolutionary synthesized circuits are compared to the known optimum size. A discussion of the potential of the information-theoretical approach is given.

In this paper, we introduce PASSSS (PAS/sup 4/), the Pareto archived and dominance selection with shrinkable search space evolutionary computation algorithm. The main contribution of this paper is a diversity control mechanism embedded into the selection operator of an evolutionary algorithm that can be used (with little or no modification) to solve both single-objective and multi-objective optimization problems. We present a detailed description of the PAS/sup 4/ algorithm, and illustrate its capabilities by solving several engineering design problems and some test functions from a well-known benchmark in evolutionary optimization. Additionally, PAS/sup 4/ is also used to solve continuous and discrete multiobjective engineering optimization problems.

This work presents a simple mechanism to handle constraints with a particle swarm optimization algorithm. Our proposal uses a simple criterion based on closeness of a particle to the feasible region in order to select a leader. Additionally, our algorithm incorporates a turbulence operator that improves the exploratory capabilities of our particle swarm optimization algorithm. Despite its relative simplicity, our comparison of results indicates that the proposed approach is highly competitive with respect to three constraint-handling techniques representative of the state-of-the-art in the area.

This paper presents an approach in which Pareto dominance is incorporated into particle swarm optimization (PSO) in order to allow this heuristic to handle problems with several objective functions. Unlike other current proposals to extend PSO to solve multiobjective optimization problems, our algorithm uses a secondary (i.e., external) repository of particles that is later used by other particles to guide their own flight. We also incorporate a special mutation operator that enriches the exploratory capabilities of our algorithm. The proposed approach is validated using several test functions and metrics taken from the standard literature on evolutionary multiobjective optimization. Results indicate that the approach is highly competitive and that can be considered a viable alternative to solve multiobjective optimization problems.

We introduce inverted and shrinkable Pareto archived evolutionary strategies, IS-PAES. This is an evolutionary algorithm for multiple objective optimization with constraint handling. IS-PAES inherits from PAES the use of an adaptable grid to keep diversity, but here this grid can grow and shrink dynamically until the constraints are met. We propose a novel approach to select a mixture of promising individuals. Several examples of the literature are used to show the potential of ISPAES.

In this paper, we propose a first version of a multi-objective evolutionary algorithm that incorporates some coevolutionary concepts. The primary design goal of the proposed approach is to reduce the total number of objective function evaluations required to produce a reasonable good approximation of the true Pareto front of a problem. The main idea of the proposed approach is to concentrate the search effort on promising regions that arise during the evolutionary process as a byproduct of a mechanism that subdivides decision variable space based on an estimate of the relative importance of each decision variable. The proposed approach is validated using several test functions taken from the specialized literature and it is compared with respect to three approaches that are representative of the state-of-the-art in evolutionary multiobjective optimization.

We present two hybrid particle swarm optimization (PSO) algorithms that incorporate a mutation operator similar to the one used with evolutionary algorithms. We study our hybridized PSO algorithm with two schemes called g/spl I.bar/best and l/spl I.bar/best, and we apply them to multimodal functions. The proposed approaches are validated using test functions taken from the specialized literature, and our results are compared with respect to those obtained by other highly competitive PSO algorithms. Our comparative study indicates that the hybridization of PSO with a nonuniform mutation operator significantly improves its performance when dealing with multimodal functions.

In this paper, we propose the use of a simple evolution strategy (SES) (i.e., a (1 + /spl lambda/)-ES with self-adaptation that uses three tournament rules based on feasibility) coupled with a diversity mechanism to solve constrained optimization problems. The proposed mechanism is based on multiobjective optimization concepts taken from an approach called the niched-Pareto genetic algorithm (NPGA). The main advantage of the proposed approach is that it does not require the definition of any extra parameters, other than those required by an evolution strategy. The performance of the proposed approach is shown to be highly competitive with respect to other constraint-handling techniques representative of the state-of-the-art in the area when using a set of well-known benchmarks.

This paper presents a simple (1 + /spl lambda/) evolution strategy and three simple selection criteria to solve engineering optimization problems. This approach avoids the use of a penalty function to deal with constraints. Its main advantage is that it does not require the definition of extra parameters, other than those used by the evolution strategy. A self-adaptation mechanism allows the algorithm to maintain diversity during the process in order to reach competitive solutions at a low computational cost. The approach was tested in four well-known engineering design problems and compared against several penalty-function-based approaches and other state-of-the-art technique. The results obtained indicate that the proposed technique is highly competitive in terms of quality, robustness and computational cost.

In this paper, we present the first proposal to use a cultural algorithm to solve multiobjective optimization problems. Our proposal uses evolutionary programming, Pareto ranking and elitism (i.e., an external population). The approach proposed is validated using several examples taken from the specialized literature. Our results are compared with respect to the NSGA-II, which is an algorithm representative of the state-of-the-art in evolutionary multiobjective optimization. The performance of our approach indicates that cultural algorithms are a viable alternative for multiobjective optimization.

We present a parallel version of a constraint-handling technique based on the artificial immune system. The proposed approach does not require penalty factors of any kind, it is relatively simple to implement and it is quite competitive with more sophisticated techniques. Additionally, when parallelized using an island scheme, the approach not only reduces its computational time, but it also improves the quality of the results produced.

This paper introduces a proposal to extend the heuristic called "particle swarm optimization" (PSO) to deal with multiobjective optimization problems. Our approach uses the concept of Pareto dominance to determine the flight direction of a particle and it maintains previously found nondominated vectors in a global repository that is later used by other particles to guide their own flight. The approach is validated using several standard test functions from the specialized literature. Our results indicate that our approach is highly competitive with current evolutionary multiobjective optimization techniques.

This paper presents a genetic programming approach for the synthesis of logic functions by means of multiplexers. The approach uses the 1-control line multiplexer as the only design unit. Any logic function (defined by a truth table) can be produced through the replication of this single unit. Our fitness function works in two stages: first, it finds feasible solutions, and then it concentrates on the minimization of the circuit, The proposed approach does not require any knowledge from the application domain.

Despite the relatively high volume of research conducted on evolutionary multiobjective optimization in the last few years. Little attention has been paid to the decision making process that is required to select a final solution to the multiobjective optimization problem at hand. This paper reviews the most important preference handling approaches used with evolutionary algorithms, analyzing their advantages and disadvantages, and then, it proposes some of the potential areas of future research in this discipline.

This paper reviews some of the most popular evolutionary multiobjective optimization techniques currently reported in the literature, indicating some of their main applications, their advantages, disadvantages, and degree of applicability. Finally, some of the most promising areas of future research are briefly discussed.

This paper introduces the notion of using coevolution to adapt the penalty factors of a fitness function incorporated in a genetic algorithm for numerical optimization. The proposed approach produces solutions even better than those previously reported in the literature for other (GA-based and mathematical programming) techniques that have been particularly fine-tuned using a normally lengthy trial and error process to solve a certain problem or set of problems. The present technique is also easy to implement and suitable for parallelization, which is a necessary further step to improve its current performance.

We describe a genetic algorithm approach to the automated design of vibratory bowl part feeders. Our approach gives us near-optimal designs in much less time than previously published optimal, brute-force search methods. We have implemented our approach in an automated part feeder design system, and we present preliminary results generated by our system.

We present a hybrid approach to optimize the counterweight balancing of a robot arm, which uses a combination of a genetic algorithm (GA) with the min-max multiobjective optimization method to get the Pareto optimal set of solutions. This set corresponds to several possible robot designs from which the most appropriate has to be chosen by the designer. Our approach is compared to a more traditional min-max search technique in which a combination of random and sequential search was used to generate the Pareto optimal solutions. Our results show how the GA is able to get solutions with a lower deviation from the ideal vector.

The paper presents a method for optimizing the design of plane and space trusses subject to a specified set of constraints. The method is based upon a search technique using genetic algorithms. Traditional structural optimization techniques consider it continuous search space, and consequently lead to unrealistic solutions because structural members are not available in continuously varying sizes. A practical method should consider only the discrete values associated with commonly available materials. On the other hand, most modern structural optimization techniques, even when they consider a discrete search space, suffer a lack of generality, and tend to be limited to a certain kind of structure. Genetic algorithms remedy these two problems since they can deal with discrete search spaces and they are general enough to be easily extended to any kind of structure without substantial modifications. Our results show the genetic algorithm can provide very good solutions, often surpassing other complex and specialized techniques.>