For black-box expensive optimization problems, the limitation in the number of function evaluations prevents evolutionary algorithms (EAs) from achieving convergence. To date, surrogate models have emerged as the predominant technique to accelerate the convergence of EAs by offering numerous virtual evaluations. However, surrogate models are often criticized for their low accuracy in fitting complex objective functions and low generalizability in handling heterogeneous decision variables. In this work, we propose an alternative idea that abandons surrogate models, focusing instead on the customization of simple EAs for expensive optimization. We construct EAs by incorporating the translation, scale, and rotation invariant variation operators, which have robust generalization capabilities due to their space independent properties, and have outstanding convergence performance due to their learnable parameterized representation. Through a series of comparative experiments, this work answers two questions: Can an EA without surrogate models outperform those with surrogate models for expensive optimization? If so, can surrogate models further accelerate the convergence of such an EA?

Sparse large-scale multi-objective optimization problems are widespread across various domains, where traditional mathematical methods and many existing multi-objective evolutionary algorithms face difficulties in achieving satisfactory results. In this paper, we propose an Adaptive Multi-Granular Pareto-optimal Subspace Learning algorithm (AMG-PSL). The algorithm employs a multi-level decision space stratification mechanism based on variable importance, implements population partitioning through k-means clustering-derived sparsity metrics, and guides population evolution using a hierarchical mutation strategy in reduced subspaces constructed by unsupervised neural networks. The algorithm incorporates a feedback-based adaptation scheme that uses offspring performance to guide solution generation and adjusts neural network architectures according to the non-dominated solutions. Experimental validation across eight benchmark problems and two practical applications demonstrates that AMG-PSL achieves superior optimization results compared to existing strategies in the domain of sparse large-scale optimization.

Non-contact voltage measurement represents a novel but challenging area within modern smart grid systems, which mainly includes structure design for sensing the electric field and voltage estimation via the sensed information. Existing approaches regarded the two components as independent optimization tasks, where the former is low-dimensional but computationally expensive while the latter is high-dimensional but relatively cheap. In this study, we argue that the non-contact measurement task is an asynchronous optimization problem with heterogeneous features in both the decision and objective spaces. Specifically, the problem includes two subproblems to be optimized sequentially, and the result of the first subproblem affects the optimum of the second one. Consequently, we propose a new optimization paradigm called evolutionary asynchronous optimization to achieve the collaborative optimization of the two tasks, and accordingly develop a new asynchronous evolutionary algorithm to solve it. Experimental results on three laboratory cases validate that, the proposed algorithm exhibits superior performance over existing multi-stage, multi-objective, and bi-level evolutionary algorithms.

Cognitive diagnosis is crucial for intelligent education because of its ability to reveal students' proficiency in knowledge concepts. Although neural network-based neural cognitive diagnosis models (CDMs) have exhibited significantly better performance than traditional models, neural cognitive diagnosis is criticized for the poor model interpretability due to the multi-layer perceptron(MLP) employed, even with the monotonicity assumption. Therefore, this paper proposes to empower the interpretability of neural cognitive diagnosis models through efficient Kolmogorov-Arnold networks (KANs), named KAN2CD, where KANs are used to enhance interpretability in two manners. Specifically, in the first manner, KANs are directly used to replace the used MLPs in existing neural CDMs; while in the second manner, the student embedding, exercise embedding, and concept embedding are directly processed by several KANs, and then their outputs are further combined and learned in a unified KAN to get final predictions. Besides, the implementation of original KANs is modified without affecting the interpretability to overcome the problem of training KANs slowly. Extensive experiments show KAN2CD outperforms traditional CDMs and slightly surpasses existing neural CDMs, and its learned structures ensure interpretability on par with traditional CDMs and better than neural CDMs. The datasets, associated code, and more experimental results are available at https://github.com/null233QAQ/KAN2CD.

Adversarial attack is a technique that introduces small and imperceptible perturbations into input data to force deep neural networks to make incorrect predictions. This not only helps assess model robustness and security but also reveals potential vulnerabilities, providing a foundation for optimizing defense mechanisms. However, single-design attack models often struggle to cope with complex and evolving defense strategies. In addition, traditional methods that rely on manual parameter tuning are inadequate for capturing internal model information in black-box scenarios, making it difficult to maintain efficiency and transferability across diverse target models and data distributions. To address this, this paper proposes a Cascade Adversarial Attack Search approach based on multi-objective optimization strategies, named CAAS. Specifically, this method constructs a comprehensive search space encompassing various attack algorithms, models, and their hyperparameter combinations. It employs a cascade strategy to sequentially apply multiple attack techniques, aiming to improve transfer attack success rates while reducing attack costs. Experimental results demonstrate that when tested on ten randomly selected models, CAAS not only significantly improves attack success rates, but also effectively controls attack costs, showcasing its superior performance in the field of adversarial attacks.

Large-scale sparse multiobjective optimization problems (LSMOPs) are of great significance in the context of practical applications, such as critical node detection, feature selection, and pattern mining. Since many LSMOPs are pursued based on large datasets, they involve a large number of decision variables, resulting in a huge search space that is challenging to explore efficiently. To rapidly approximate sparse Pareto optimal solutions, some evolutionary algorithms have been proposed to reduce the dimensionality of LSMOPs. However, their adaptability to different LSMOPs remains limited due to their reliance on fixed dimensionality reduction schemes, which can potentially lead to local optima and inefficient utilization of function evaluations. To address this issue, a knowledge learning-based dimensionality reduction approach is proposed in this article. First, in the early stages of evolution, the impact of different dimensionality reduction schemes on the sparse distribution of the population is evaluated. Then, the multilayer perceptron is employed to learn the accumulated knowledge from the evolutionary process, thereby constructing a mapping model between the sparse features of the evolutionary process and the candidate dimensionality reduction schemes. Finally, the model recommends the best dimensionality reduction scheme in each generation, achieving a good balance between exploration and exploitation. Experimental evaluations on both benchmark and real-world LSMOPs demonstrate that an evolutionary algorithm incorporating the proposed knowledge learning-based dimensionality reduction approach outperforms most existing evolutionary algorithms.

Robust multiobjective optimization problems (RMOPs) widely exist in real-world applications, which introduce a variety of uncertainty in optimization models. While some evolutionary algorithms have been developed to find optimal solutions robust to uncertainty, they are ineffective to handle RMOPs in high-dimensional decision spaces. Focusing on the large-scale RMOPs with sparse optimal solutions, this article proposes an evolutionary algorithm with novel strategies for the selection, generation, and evaluation of robust solutions. In order to handle the uncertainty in the optimization models, we first introduce an archive to separately consider optimality and robustness, which can achieve the selection of robust solutions effectively at a low cost. Based on the robust knowledge extracted from the archive, a guiding vector is adaptively updated to facilitate the generation of robust solutions in high-dimensional decision spaces. With the assistance of the guiding vector, a robustness indicator is suggested to assist in the evaluation of robust solutions without additional perturbations. Besides, we design a test suite to evaluate the performance of the proposed algorithm on the large-scale RMOPs. Our experimental results demonstrate that the proposed algorithm has significant advantages over the state-of-the-art evolutionary algorithms in terms of optimality and robustness, on both the proposed test suite and practical applications.

Despite that the surrogate-assisted evolutionary algorithms have achieved great success in addressing expensive optimization problems, they still suffer from stiff challenges when the number of dimensions of problems becomes large. The primary reason lies in that it is very hard to build an acceptable surrogate model in the high-dimensional search space with small amounts of evaluated historical data in evolution. To tackle this issue, we suggest an effective surrogate modeling method for large-scale expensive optimization in this paper, where the models are built on a number of linear subspaces instead of the original search space. Specifically, a linear subspace is constructed by a pair of points/solutions which are generated based on the set of elite solutions. For each linear subspace, several historical solutions are first associated according to their distance to the linear subspace, and then a surrogate model is trained by the associated solutions and used to evaluate the offspring. To ensure the exploration and exploitation capacity of the proposed method, these linear subspaces and the surrogate models are updated after a few iterations. Experimental results on CEC’2010 and CEC’2013 single-objective optimization problems with up to 1500 decision variables show that the proposed algorithm is superior over six comparison algorithms. Moreover, we also extend the proposed algorithm to multi-objective optimization problems and verified its competitiveness on problems with up to 1500 decision variables.

Binary optimization assumes a pervasive significance in the context of practical applications, such as knapsack problems, maximum cut problems, and critical node detection problems. Existing techniques including mathematical programming, heuristics, evolutionary computation, and neural networks have been employed to tackle binary optimization problems (BOPs), however, they grapple with the challenge of optimizing a large number of binary variables. In this paper, we propose a dimensionality reduction method to assist evolutionary algorithms in solving large-scale BOPs, which is achieved based on neural networks. The proposed method converts the optimization of a large number of binary variables into the optimization of a small number of network weights, resulting in a significant reduction in search space dimensionality. Crucially, the proposed method obviates the necessity for a training process, which eliminates the requirement for a priori knowledge and enhances the search efficiency. On six types of single-and multi-objective BOPs with up to 10 000 000 variables, the proposed method demonstrates superiority over top-tier evolutionary algorithms and neural network-based methods.

Multitasking multiobjective evolutionary algorithms (MMEAs) have been extensively studied in the past decade, which mainly concentrate on multitasking multiobjective optimization problems (MMOPs) with dozens of decision variables. Nevertheless, many real-world MMOPs have thousands of decision variables and are of sparse nature, which are regarded as large-scale MMOPs (LSMMOPs) in this study. To address LSMMOPs, a sparsity knowledge transfer-based evolutionary multiobjective algorithm, termed EMO-SKT, is proposed for efficiently finding high-quality sparse solutions of LSMMOPs. For each target optimization task, a sparsity knowledge transfer strategy extracts sparse distribution information from a source task and incorporates the information into the target task for two types of sparsity knowledge: 1) variable importance and 2) sparse degree. The variable importance is utilized to produce high-quality sparse solutions during the evolutionary search for EMO-SKT, while the sparse degree facilitates the reduction of search space and thus speeds up the convergence of the evolutionary search. Experimental results on eight benchmark problems and six practical LSMMOPs demonstrate the effectiveness of the sparsity knowledge transfer strategy. Furthermore, the proposed EMO-SKT is capable of efficiently finding high-quality sparse solutions on an LSMMOP with over 1000 decision variables. In comparison with five state-of-the-art multitasking or sparse optimization algorithms, the proposed EMO-SKT exhibits superior performance in terms of both solution quality and search efficiency.

As a pivotal technique in intelligent education systems, cognitive diagnosis (CD) serves to reveal students’ knowledge proficiency for better tackling subsequent tasks. Unfortunately, due to pursuing high model interpretability, existing manually designed models for CD often hold simplistic architectures, which cannot cope with intricate data in modern education platforms. Furthermore, the bias of human design limits the emergence of novel and effective CD models (CDMs). To develop interpretable and more effective models, thus this article proposes an evolutionary multiobjective neural architecture search (NAS) approach for CD. Specifically, we first adopt a comprehensive search space for the NAS task of CD: all candidate models can be encompassed by a general model that deals with three distinct types of inputs. Then, an innovative model interpretability objective is devised to formulate the architecture search task as a bi-objective optimization problem (BOP). To solve the BOP, we employ a multiobjective genetic programming (MOGP) as the search strategy to explore the search space. To make the employed MOGP search well, all architectures are first encoded by trees for easy optimization, and we devise a genetic operation and a population initialization strategy to expedite its convergence. Finally, the proposed approach is actually an MOGP-based NAS approach for CD. Extensive experiments show that CDMs searched by the proposed approach exhibit significantly better performance than existing models and hold as good interpretability as handcrafted models. Besides, the effectiveness of the proposed MOGP search strategy, the devised objective, and tailored strategies are validated.

Evolutionary algorithms have demonstrated superior performance in solving multi-objective optimization problems (MOPs), but no single algorithm is consistently effective across all MOPs. When using evolutionary algorithms to solve MOPs, environmental selection strategies determining which solutions should survive are crucial to population evolution. While different environmental selection strategies exhibit different search behaviors on various MOPs, existing multi-objective evolutionary algorithms rarely focus on the adaptation of environmental selection strategies. To fill this gap, this paper proposes a framework for assembling environmental selection strategies, which utilizes neural networks to assess the effects of different strategies on population evolution, and employs reinforcement learning to adaptively select the most effective strategies. The effectiveness and versatility of the proposed framework are verified on four test sets, where the proposed framework shows significant superiority over the state-of-the-art.

The traveling salesman problem (TSP) has long been a central focus in combinatorial optimization, with extensive applications in areas such as logistics and transportation. With the rapid development of machine learning, the research focusing on TSPs has shifted from traditional heuristics, often involving a substantial amount of manually designed rules, to machine learning-based approaches. However, due to the limitations of models and hardware, generative approaches based on machine learning often fall short of expectations in terms of effectiveness, and they generally lag behind well-established traditional heuristics. To address these limitations and leverage the strengths of both approaches, the integration of heuristics with machine learning has become a crucial branch for solving TSPs. Thus, this paper proposes an approach that combines a traditional heuristic, specifically guided fast local search, with a neural network to tackle TSPs, where reinforcement learning techniques are employed to train the neural network. Experimental results showcase the superiority of the proposed approach over the original heuristic, outperforming the majority of mainstream machine learning-based approaches. Notably, the proposed approach achieves zero-gap results for instances with 20 and 50 nodes and a minimal gap of 0.07% for instances with 100 nodes. Compared with existing reinforcement learning-based approaches, the proposed approach also stands out with the best overall performance.

In the face of a series of challenging control tasks, multi-agent reinforcement learning has demonstrated its superiority. However, it suffers from certain drawbacks like deceptive reward functions, unstable training processes, and a lack of exploration for novel policies. To address these issues, the combination of evolutionary algorithms and reinforcement learning has been introduced, as evolutionary algorithms possess good exploration and convergence properties, enabling reinforcement learning to better leverage its performance. To enhance the complementary advantages of evolutionary algorithms and reinforcement learning, we combine them into an asymmetric reinforcement learning framework by utilizing a shared observation encoder, allowing the algorithm's performance to break through certain bottlenecks. In addition to the consideration of reward, we propose the concept of policy novelty, which measures the degree of difference between a policy's behavior and others. Consequently, the training of multiple agents is formulated as a large-scale bi-objective optimization problem, and solved by a multi-objective evolutionary algorithm. The proposed approach is tested on eight experimental tasks within the MA-MuJoCo framework, exhibiting superiority over commonly used approaches.

The purpose of this study was to develop an evolutionary algorithm (EA) with bilevel surrogate modeling, called BL-SAEA, for tackling bilevel optimization problems (BLOPs), in which an upper level problem is to be solved subject to the optimality of a corresponding lower level problem. The motivation of this article is that the extensive lower level optimization required by each upper level solution consumes too many function evaluations, leading to poor optimization performance of EAs. To this end, during the upper level optimization, the BL-SAEA builds an upper level surrogate model to select several promising upper level solutions for the lower level optimization. Because only a small number of upper level solutions require the lower level optimization, the number of function evaluations can be considerably reduced. During the lower level optimization, the BL-SAEA constructs multiple lower level surrogate models to initialize the population of the lower level optimization, thus further decreasing the number of function evaluations. Experimental results on two widely used benchmarks and two real-world BLOPs demonstrate the superiority of our proposed algorithm over six state-of-the-art algorithms in terms of effectiveness and efficiency.

Multiobjective optimization problems (MOPs) with expensive constraints pose stiff challenges to existing surrogate-assisted evolutionary algorithms (SAEAs) in a very limited computational cost, due to the fact that the number of expensive constraints for an MOP is often large. For existing SAEAs, they always approximate constraint functions in a single granularity, namely, approximating the constraint violation (CV, coarse-grained) or each constraint (fine-grained). However, the landscape of CV is often too complex to be accurately approximated by a surrogate model. Although the modeling of each constraint function may be simpler than that of CV, approximating all the constraint functions independently may result in tremendous cumulative errors and high computational costs. To address this issue, in this article, we develop a multigranularity surrogate modeling framework for evolutionary algorithms (EAs), where the approximation granularity of constraint surrogates is adaptively determined by the position of the population in the fitness landscape. Moreover, a dedicated model management strategy is also developed to reduce the impact resulting from the errors introduced by constraint surrogates and prevent the population from trapping into local optima. To evaluate the performance of the proposed framework, an implementation called K-MGSAEA is proposed, and the experimental results on a large number of test problems show that the proposed framework is superior to seven state-of-the-art competitors.

The pneumonia caused by COVID-19 is spreading worldwide, threatening human health and life. In the last three years, China has taken a series of effective measures to prevent the spread of the virus, where a core measure is the stay-home quarantine imposed in infected communities. The quarantine can effectively reduce physical contacts and transmission risk, however, it encounters several difficulties especially the guarantee of living materials. In order to ensure the quantity and freshness of living materials such as vegetables and fruits, it is necessary to construct multi-level logistics distribution centers, where the selection of locations for these centers becomes a vital issue. Such facility location problems are challenging in terms of both modeling and optimization, especially when facing the millions of residents and thousands of communities in a city commonly existing in China. In this study, we build a large-scale multi-objective optimization model for the distribution center location problem, and solve it via state-of-the-art sparse evolutionary algorithms. The experimental results verify that the center locations obtained by our approach can save human resources while reducing the risk of virus propagation.

Sparse large-scale multi-objective optimization problems (LSMOPs) widely exist in real-world applications. Due to the characteristics of high-dimensional search space and sparse Pareto optimal solutions, most existing multi-objective evolutionary algorithms encounter difficulties in solving this kind of optimization problems. In this paper, a sparsity-guided elitism co-evolutionary framework, namely SGECF, is proposed. At each generation, SGECF first performs k-means clustering on non-dominated solutions to calculate the present best sparsity. Then, SGECF divides the population into a winner subpopulation and two loser subpopulations based on the Pareto dominance relationship and the present best sparsity. In the process of offspring reproduction, the winner subpopulation generates offspring solutions using conventional genetic operators, and the two loser subpopulations reproduce offspring solutions under the guidance of the present best sparsity and decision variable importance. In the experiments, the performance of SGECF is investigated on eight benchmark problems and a real-world application. Experimental results show that SGECF is superior over four state-of-the-art algorithms.

For large-scale multi-objective evolutionary algorithms (LSMOEAs), it has been a major challenge to efficiently obtain accurate evolutionary directions in the ultra-high-dimensional decision space to produce high-quality offspring. To this hand, this paper proposes an algorithm, namely, a directed quick search guided large-scale multi-objective evolutionary algorithm (QSLMOA). This algorithm contains three main parts: a directional vector-based sampling strategy, a quick search guided directed reproduction strategy, and an environment selection. In each generation, the proposed sampling strategy determines a set of directional solutions to construct the direction vectors which are used to guide the search directions. The sampling strategy significantly reduces the search space and improves the sampling efficiency in the early stages. On the other hand, our proposed reproduction strategy introduces the directional information, and with their assistance, the inferior ones of solutions in the combined population can rapidly converge to the elite ones, which can speed up the convergence rate. Finally, the elitist non-dominated sorting is adopted as the environment selection to obtain the parent population of the next generation. Comprehensive experiments verify that QSLMOA performs the best compared to the state-of-the-art LSMOEAs for nine large-scale multi-objective benchmark problems LSMOP1-LSMOP9 with up to three objectives and 5000 decision variables.

Current model quantization methods have shown their promising capability in reducing storage space and computation complexity. However, due to the diversity of quantization forms supported by different hardware, one limitation of existing solutions is that usually require repeated optimization for different scenarios. How to construct a model with flexible quantization forms has been less studied. In this paper, we explore a one-shot network quantization regime, named Elastic Quantization Neural Networks (EQ-Net), which aims to train a robust weight-sharing quantization supernet. First of all, we propose an elastic quantization space (including elastic bit-width, granularity, and symmetry) to adapt to various mainstream quantitative forms. Secondly, we propose the Weight Distribution Regularization Loss (WDR-Loss) and Group Progressive Guidance Loss (GPG-Loss) to bridge the inconsistency of the distribution for weights and output logits in the elastic quantization space gap. Lastly, we incorporate genetic algorithms and the proposed Conditional Quantization-Aware Accuracy Predictor (CQAP) as an estimator to quickly search mixed-precision quantized neural networks in supernet. Extensive experiments demonstrate that our EQ-Net is close to or even better than its static counterparts as well as state-of-the-art robust bit-width methods. Code can be available at https://github.com/xuke225/EQ-Net.

Knowledge tracing (KT) aims to trace students' knowledge states by predicting whether students answer correctly on exercises. Despite the excellent performance of existing Transformer-based KT approaches, they are criticized for the manually selected input features for fusion and the defect of single global context modelling to directly capture students' forgetting behavior in KT, when the related records are distant from the current record in terms of time. To address the issues, this paper first considers adding convolution operations to the Transformer to enhance its local context modelling ability used for students' forgetting behavior, then proposes an evolutionary neural architecture search approach to automate the input feature selection and automatically determine where to apply which operation for achieving the balancing of the local/global context modelling. In the search space, the original global path containing the attention module in Transformer is replaced with the sum of a global path and a local path that could contain different convolutions, and the selection of input features is also considered. To search the best architecture, we employ an effective evolutionary algorithm to explore the search space and also suggest a search space reduction strategy to accelerate the convergence of the algorithm. Experimental results on the two largest and most challenging education datasets demonstrate the effectiveness of the architecture found by the proposed approach.

Most existing evolutionary search strategies are not so efficient when directly handling the decision space of large-scale multiobjective optimization problems (LMOPs). To enhance the efficiency of tackling LMOPs, this article proposes an accelerated evolutionary search (AES) strategy. Its main idea is to learn a gradient-descent-like direction vector (GDV) for each solution via the specially trained feedforward neural network, which may be the learnt possibly fastest convergent direction to reproduce new solutions efficiently. To be specific, a multilayer perceptron (MLP) with only one hidden layer is constructed, in which the number of neurons in the input and output layers is equal to the dimension of the decision space. Then, to get appropriate training data for the model, the current population is divided into two subsets based on the nondominated sorting, and each poor solution in one subset with worse convergence will be paired to an elitist solution in another subset with the minimum angle to it, which is considered most likely to guide it with rapid convergence. Next, this MLP is updated via backpropagation with gradient descent by using the above elaborately prepared dataset. Finally, an accelerated large-scale multiobjective evolutionary algorithm (ALMOEA) is designed by using AES as a reproduction operator. Experimental studies validate the effectiveness of the proposed AES when handling the search space of LMOPs with dimensionality ranging from 1000 to 10000. When compared with six state-of-the-art evolutionary algorithms, the experimental results also show the better efficiency and performance of the proposed optimizer in solving various LMOPs.

The Pareto front (PF) estimation has become an emerging strategy for solving multiobjective optimization problems in recent studies. By approximating the geometrical structure of the PF during the evolutionary procedure, some PF estimation approaches have been suggested and shown effectiveness in guiding the search direction of evolutionary algorithms. However, these approaches encounter difficulties in handling irregular PFs, whose geometrical structures are too complex to be properly approximated. To address this issue, this article proposes a novel PF estimation approach based on local models. In contrast to existing approaches estimating the PF via a reference point set or a single model, the proposed approach automatically divides the population into several groups and builds a local model for each group of solutions. In spite of the simplicity of each local model, the combination of all the local models can approximate the PFs with complex geometrical structures. An evolutionary algorithm is then developed based on the local model-based PF estimation approach and a novel fitness function and is compared with four evolutionary algorithms on 39 problems. Statistical results indicate that the proposed algorithm exhibits better performance than the compared algorithms, especially on problems with highly irregular PFs.

Evolutionary algorithms and mathematical programming methods are currently the most popular optimizers for solving continuous optimization problems. Owing to the population based search strategies, evolutionary algorithms can find a set of promising solutions without using any problem-specific information. By contrast, with the assistance of gradient and other information of the functions, mathematical programming methods can quickly converge to a single optimum. While these two types of optimizers have their own advantages and disadvantages, the performance comparison between them is rarely touched. It is known that gradient descent methods generally converge faster than evolutionary algorithms, but when can evolutionary algorithms outperform gradient descent methods? How is the scalability of them? To answer these questions, this paper first gives a review of popular evolutionary algorithms and mathematical programming methods, then conducts several experiments to compare their performance from various aspects, and finally draws some conclusions.

A smart recommendation method can greatly improve tourists' travel experience, and it is an important task for tourism recommendation systems to intelligently suggest scenic spots for tourists according to their historical visit records. Currently, the collaborative filtering and deep neural network-based methods occupy the mainstream of tourism recommendation systems. Although each type of recommendation methods is superior over the others in terms of different aspects, the performance of a single recommendation method is limited. In order to inherit the advantages of different types of recommendation methods, this work suggests a hybrid method for assembling multiple methods for tourism recommendation. Based on the scenic spots obtained by multiple recommendation methods, the proposed hybrid method uses two novel objectives to evaluate each scenic spot, and identifies the best$K$scenic spots via the techniques used in evolutionary multi-objective optimization. In comparison to existing recommendation methods and hybrid methods, the proposed hybrid method exhibits better performance on two public tourism datasets and a new dataset created based on the tourism information of Huangshan City.

In real-world applications, there exist a lot of multiobjective optimization problems whose Pareto-optimal solutions are sparse, that is, most variables of these solutions are 0. Generally, many sparse multiobjective optimization problems (SMOPs) contain a large number of variables, which pose grand challenges for evolutionary algorithms to find the optimal solutions efficiently. To address the curse of dimensionality, this article proposes an evolutionary algorithm for solving large-scale SMOPs, which aims to mine the sparse distribution of the Pareto-optimal solutions and, thus, considerably reduces the search space. More specifically, the proposed algorithm suggests an evolutionary pattern mining approach to detect the maximum and minimum candidate sets of the nonzero variables in the Pareto-optimal solutions, and uses them to limit the dimensions in generating offspring solutions. For further performance enhancement, a binary crossover operator and a binary mutation operator are designed to ensure the sparsity of solutions. According to the results on eight benchmark problems and four real-world problems, the proposed algorithm is superior over existing evolutionary algorithms in solving large-scale SMOPs.

Large-scale multiobjective optimization problems (LMOPs) bring significant challenges for traditional evolutionary operators, as their search capability cannot efficiently handle the huge decision space. Some newly designed search methods for LMOPs usually classify all variables into different groups and then optimize the variables in the same group with the same manner, which can speed up the population's convergence. Following this research direction, this article suggests a differential evolution (DE) algorithm that favors searching the variables with higher importance to the solving of LMOPs. The importance of each variable to the target LMOP is quantized and then all variables are categorized into different groups based on their importance. The variable groups with higher importance are allocated with more computational resources using DE. In this way, the proposed method can efficiently generate offspring in a low-dimensional search subspace formed by more important variables, which can significantly speed up the convergence. During the evolutionary process, this search subspace for DE will be expanded gradually, which can strike a good balance between exploration and exploitation in tackling LMOPs. Finally, the experiments validate that our proposed algorithm can perform better than several state-of-the-art evolutionary algorithms for solving various benchmark LMOPs.

Both objective optimization and constraint satisfaction are crucial for solving constrained multiobjective optimization problems, but the existing evolutionary algorithms encounter difficulties in striking a good balance between them when tackling complex feasible regions. To address this issue, this article proposes a two-stage evolutionary algorithm, which adjusts the fitness evaluation strategies during the evolutionary process to adaptively balance objective optimization and constraint satisfaction. The proposed algorithm can switch between the two stages according to the status of the current population, enabling the population to cross the infeasible region and reach the feasible regions in one stage, and to spread along the feasible boundaries in the other stage. Experimental studies on four benchmark suites and three real-world applications demonstrate the superiority of the proposed algorithm over the state-of-the-art algorithms, especially on problems with complex feasible regions.

Intensity-modulated radiotherapy (IMRT) is one of the most popular techniques for cancer treatment. However, existing IMRT planning methods can only generate one solution at a time and, consequently, medical physicists should perform the planning process many times to obtain diverse solutions to meet the requirement of a clinical case. Meanwhile, multiobjective evolutionary algorithms (MOEAs) have not been fully exploited in IMRT planning since they are ineffective in optimizing the large number of discrete variables of IMRT. To bridge the gap, this article formulates IMRT planning into a large-scale combinatorial many-objective optimization problem and proposes a coevolutionary algorithm to solve it. In contrast to the existing MOEAs handling high-dimensional search spaces via variable grouping or dimensionality reduction, the proposed algorithm evolves one population with fine encoding for local exploitation and evolves another population with rough encoding for global exploration. Moreover, the convergence speed is further accelerated by two customized local search strategies. The experimental results verify that the proposed algorithm outperforms state-of-the-art MOEAs and IMRT planning methods on a variety of clinical cases.

Due to natural disasters or system failures, the facility has the risk of disruption, and thus improving the location reliability under uncertainty of facilities becomes an important issue. In this paper, we propose a multi-objective facility location problem under uncertainty of facilities, where two objectives on reliability are constructed and multiple coverage with variable radius is imposed to reduce the influence caused by the facility disruption. A dual-population based evolutionary algorithm is also suggested to address this problem, where one population is for the location optimization and the other population is for the radius optimization. These two populations iteratively exchange the information obtained from elite solutions during the evolution to collaboratively search for the optimal solutions of the problem. The location population provides the high-quality location schemes for radius population in evaluating the quality of radii of each location, whereas the radius population equips the proper radii for location population in determining the good location schemes. Experimental results indicate that the proposed model can effectively improve the location reliability and the proposed method can obtain higher quality optimal solutions in comparison with four state-of-the-art algorithms. Moreover, the proposed method is applied to a real-world facility location with uncertainty of express cabinets in Tianjin, China, and it produces satisfactory location schemes.

Dynamic vehicle routing problems (DVRPs) have become a hot research topic due to their significance in logistics, although it is still very challenging for existing algorithms to solve DVRPs due to the dynamically changing customer requests during the optimization. In this paper, we propose a pairwise proximity learning-based ant colony algorithm, termed PPL-ACO, for tackling DVRPs. In PPL-ACO, a pairwise proximity learning method is suggested to predict the local visiting order of customers in the optimal route after the occurrence of changes, which is on the basis of learning from the optimal routes found before the changes occur. A radial basis function network is used to learn the local visiting order of customers based on the proximity between each pair of customer nodes, by which the optimal routes can be quickly tracked after changes occur. Experimental results on 22 popular DVRP instances show that the proposed PPL-ACO significantly outperforms four state-of-the-art approaches to DVRPs. More interestingly, the results on five large-scale DVRP instances demonstrate the superiority of the proposed PPL-ACO in solving large-scale DVPRs with up to 1000 customers. The results on a real case of Nankai Strict, Tianjin, China also verifies that the proposed PPL-ACO is more effective and efficient than the four compared approaches in solving real-world DVRPs.

It has been widely recognized that the efficient training of neural networks (NNs) is crucial to classification performance. While a series of gradient-based approaches have been extensively developed, they are criticized for the ease of trapping into local optima and sensitivity to hyperparameters. Due to the high robustness and wide applicability, evolutionary algorithms (EAs) have been regarded as a promising alternative for training NNs in recent years. However, EAs suffer from the curse of dimensionality and are inefficient in training deep NNs (DNNs). By inheriting the advantages of both the gradient-based approaches and EAs, this article proposes a gradient-guided evolutionary approach to train DNNs. The proposed approach suggests a novel genetic operator to optimize the weights in the search space, where the search direction is determined by the gradient of weights. Moreover, the network sparsity is considered in the proposed approach, which highly reduces the network complexity and alleviates overfitting. Experimental results on single-layer NNs, deep-layer NNs, recurrent NNs, and convolutional NNs (CNNs) demonstrate the effectiveness of the proposed approach. In short, this work not only introduces a novel approach for training DNNs but also enhances the performance of EAs in solving large-scale optimization problems.

Due to the curse of dimensionality of search space, it is extremely difficult for evolutionary algorithms to approximate the optimal solutions of large-scale multiobjective optimization problems (LMOPs) by using a limited budget of evaluations. If the Pareto-optimal subspace is approximated during the evolutionary process, the search space can be reduced and the difficulty encountered by evolutionary algorithms can be highly alleviated. Following the above idea, this article proposes an evolutionary algorithm to solve sparse LMOPs by learning the Pareto-optimal subspace. The proposed algorithm uses two unsupervised neural networks, a restricted Boltzmann machine, and a denoising autoencoder to learn a sparse distribution and a compact representation of the decision variables, where the combination of the learnt sparse distribution and compact representation is regarded as an approximation of the Pareto-optimal subspace. The genetic operators are conducted in the learnt subspace, and the resultant offspring solutions then can be mapped back to the original search space by the two neural networks. According to the experimental results on eight benchmark problems and eight real-world problems, the proposed algorithm can effectively solve sparse LMOPs with 10000 decision variables by only 100000 evaluations.

Constrained multiobjective optimization problems (CMOPs) widely exist in real-world applications, and they are challenging for conventional evolutionary algorithms (EAs) due to the existence of multiple constraints and objectives. When the number of objectives or decision variables is scaled up in CMOPs, the performance of EAs may degenerate dramatically and may fail to obtain any feasible solutions. To address this issue, we propose a paired offspring generation-based multiobjective EA for constrained large-scale optimization. The general idea is to emphasize the role of offspring generation in reproducing some promising feasible or useful infeasible offspring solutions. We first adopt a small set of reference vectors for constructing several subpopulations with a fixed number of neighborhood solutions. Then, a pairing strategy is adopted to determine some pairwise parent solutions for offspring generation. Consequently, the pairwise parent solutions, which could be infeasible, may guide the generation of well-converged solutions to cross the infeasible region(s) effectively. The proposed algorithm is evaluated on CMOPs with up to 1000 decision variables and ten objectives. Moreover, each component in the proposed algorithm is examined in terms of its effect on the overall algorithmic performance. Experimental results on a variety of existing and our tailored test problems demonstrate the effectiveness of the proposed algorithm in constrained large-scale multiobjective optimization.

Multimodal multiobjective optimization problems (MMOPs) widely exist in real-world applications, which have multiple equivalent Pareto-optimal solutions that are similar in the objective space but totally different in the decision space. While some evolutionary algorithms (EAs) have been developed to find the equivalent Pareto-optimal solutions in recent years, they are ineffective to handle large-scale MMOPs having a large number of variables. This article thus proposes an EA for solving large-scale MMOPs with sparse Pareto-optimal solutions, i.e., most variables in the optimal solutions are 0. The proposed algorithm explores different regions of the decision space via multiple subpopulations and guides the search behavior of the subpopulations via adaptively updated guiding vectors. The guiding vector for each subpopulation not only provides efficient convergence in the huge search space but also differentiates its search direction from others to handle the multimodality. While most existing EAs solve MMOPs with 2-7 decision variables, the proposed algorithm is shown to be effective for benchmark MMOPs with up to 500 decision variables. Moreover, the proposed algorithm also produces a better result than state-of-the-art methods for the neural architecture search.

Constrained multiobjective optimization problems (CMOPs) are challenging because of the difficulty in handling both multiple objectives and constraints. While some evolutionary algorithms have demonstrated high performance on most CMOPs, they exhibit bad convergence or diversity performance on CMOPs with small feasible regions. To remedy this issue, this article proposes a coevolutionary framework for constrained multiobjective optimization, which solves a complex CMOP assisted by a simple helper problem. The proposed framework evolves one population to solve the original CMOP and evolves another population to solve a helper problem derived from the original one. While the two populations are evolved by the same optimizer separately, the assistance in solving the original CMOP is achieved by sharing useful information between the two populations. In the experiments, the proposed framework is compared to several state-of-the-art algorithms tailored for CMOPs. High competitiveness of the proposed framework is demonstrated by applying it to 47 benchmark CMOPs and the vehicle routing problem with time windows.

The algorithm recommendation is attracting increasing attention in solving real-world capacitated vehicle routing problems (CVRPs), due to the fact that existing meta-heuristic algorithms often show different performances on different CVRPs. To effectively perform algorithm recommendation for CVRPs, it becomes vital to extract suitable features to characterize the CVRPs accurately. To this end, in this article three groups of penetrating features are proposed to capture the characteristics of CVRPs. The first group consists of some basic features of CVRPs, where several features are suggested to capture the distribution of customer demand, the relationship between customer demand and vehicle capacity, besides some common attributes widely used in CVRPs. The second group is composed of the features extracted from some CVRP solutions generated by local search, where in addition to the feasible and better solutions, the worse solutions and the distribution of travel cost are also used to measure the sensitivity of CVRPs to local search operations. The third group is made up of image features obtained by depicting CVRP instances through images, which is first introduced by us to enhance the generalization of algorithm recommendation. Furthermore, based on the three groups of features, an algorithm recommendation method called ARM-I is built on the basis of a KNN classifier to recommend suitable algorithm for CVRPs. Experimental results on several selected benchmarks demonstrate the effectiveness of the designed features. More interestingly, the proposed ARM-I shows high generalization on real-world instances.

Diversity preservation is a crucial technique in multiobjective evolutionary algorithms (MOEAs), which aims at evolving the population toward the Pareto front (PF) with a uniform distribution and a good extensity. In spite of many diversity preservation approaches in existing MOEAs, most of them encounter difficulties in tackling complex PFs. This article gives a detail introduction to existing diversity preservation approaches, as well as a revelation of the limitations of them. To address the limitations of existing diversity preservation approaches, this article proposes a multistage MOEA for better diversity performance. The proposed MOEA divides the optimization process into multiple stages according to the population in each generation, and updates the population by different steady-state selection schemes in different stages. According to the experimental results on 21 benchmark problems, the proposed MOEA exhibits better diversity performance than 11 existing MOEAs.

Due to the curse of dimensionality, two main issues remain challenging for applying evolutionary algorithms (EAs) to large-scale multiobjective optimization. The first issue is how to improve the efficiency of EAs for reducing computation cost. The second one is how to improve the diversity maintenance of EAs to avoid local optima. Nevertheless, these two issues are somehow conflicting with each other, and thus it is crucial to strike a balance between them in practice. Thereby, we propose an iterated problem reformulation based EA for large-scale multiobjective optimization, where the problem reformulation based method and the decomposition based method are used iteratively to address the aforementioned issues. The proposed method is compared with several state-of-the-art EAs on a variety of large-scale multiobjective optimization problems. Experimental results demonstrate the effectiveness of our proposed iterated method in large-scale multiobjective optimization.

It has been widely recognized that evolutionary computation is one of the most effective techniques for solving complex optimization problems. As a group of meta-heuristics inspired by nature, the superiority of evolutionary algorithms is mainly attributed to the evolution of multiple candidate solutions, which can strike a balance between exploration and exploitation. However, the effectiveness of evolutionary algorithms is generally at the expense of efficiency, which reduces the prevalence of evolutionary algorithms in solving real-world optimization problems. In 2017, we proposed the evolutionary multi-objective optimization platform PlatEMO to facilitate the use of multi-objective evolutionary algorithms (MOEAs), where some delicate techniques were developed to improve the computational efficiency of MOEAs. These techniques have not been introduced before, since users need not care about them when using existing MOEAs or developing new MOEAs. To deepen the understanding of the core mechanisms of PlatEMO, this paper gives a comprehensive introduction to these techniques, including new non-dominated sorting approaches, matrix calculation, and parallel computing. Several comparative experiments are conducted for a quantitative understanding of the efficiency improvement brought by these techniques.

In many-objective optimization problems (MaOPs), forming sound tradeoffs between convergence and diversity for the environmental selection of evolutionary algorithms is a laborious task. In particular, strengthening the selection pressure of population toward the Pareto-optimal front becomes more challenging, since the proportion of nondominated solutions in the population scales up sharply with the increase of the number of objectives. To address these issues, this paper first defines the nondominated solutions exhibiting evident tendencies toward the Pareto-optimal front as prominent solutions, using the hyperplane formed by their neighboring solutions, to further distinguish among nondominated solutions. Then, a novel environmental selection strategy is proposed with two criteria in mind: 1) if the number of nondominated solutions is larger than the population size, all the prominent solutions are first identified to strengthen the selection pressure. Subsequently, a part of the other nondominated solutions are selected to balance convergence and diversity and 2) otherwise, all the nondominated solutions are selected; then a part of the dominated solutions are selected according to the predefined reference vectors. Moreover, based on the definition of prominent solutions and the new selection strategy, we propose a hyperplane assisted evolutionary algorithm, referred here as hpaEA, for solving MaOPs. To demonstrate the performance of hpaEA, extensive experiments are conducted to compare it with five state-of-the-art many-objective evolutionary algorithms on 36 many-objective benchmark instances. The experimental results show the superiority of hpaEA which significantly outperforms the compared algorithms on 20 out of 36 benchmark instances.

In evolutionary multiobjective optimization, the Pareto front (PF) is approximated by using a set of representative candidate solutions with good convergence and diversity. However, most existing multiobjective evolutionary algorithms (MOEAs) have general difficulty in the approximation of PFs with complicated geometries. To address this issue, we propose a generic front modeling method for evolutionary multiobjective optimization, where the shape of the nondominated front is estimated by training a generalized simplex model. On the basis of the estimated front, we further develop an MOEA, where both the mating selection and environmental selection are driven by the approximate nondominated fronts modeled during the optimization process. For performance assessment, the proposed algorithm is compared with several state-of-the-art evolutionary algorithms on a wide range of benchmark problems with various types of PFs and different numbers of objectives. Experimental results demonstrate that the proposed algorithm performs consistently on a variety of multiobjective optimization problems.

There exist many multiobjective optimization problems (MOPs) containing a large number of decision variables in real-world applications, which are known as large-scale MOPs. Due to the ineffectiveness of existing operators in finding optimal solutions in a huge decision space, some decision variable division-based algorithms have been tailored for improving the search efficiency in solving large-scale MOPs. However, these algorithms will encounter difficulties when solving problems with complicated landscapes, as the decision variable division is likely to be inaccurate and time consuming. In this paper, we propose a competitive swarm optimizer (CSO)-based efficient search for solving large-scale MOPs. The proposed algorithm adopts a new particle updating strategy that suggests a two-stage strategy to update position, which can highly improve the search efficiency. The experimental results on large-scale benchmark MOPs and an application example demonstrate the superiority of the proposed algorithm over several state-of-the-art multiobjective evolutionary algorithms, including problem transformation-based algorithm, decision variable clustering-based algorithm, particle swarm optimization algorithm, and estimation of distribution algorithm.

Ratio error (RE) estimation of the voltage transformers (VTs) plays an important role in modern power delivery systems. Existing RE estimation methods mainly focus on periodical calibration but ignore the time-varying property. Consequently, it is difficult to efficiently estimate the state of the VTs in real time. To address this issue, we formulate a time-varying RE estimation (TREE) problem into a large-scale multiobjective optimization problem, where the multiple objectives and inequality constraints are formulated by statistical and physical rules extracted from the power delivery systems. Furthermore, a set of TREE problems from different substations is systematically formulated into a benchmark test suite for characterizing their different properties. The formulation of these TREE problems not only transfers an expensive RE estimation task to a relatively cheaper optimization problem but also promotes the research in large-scale multiobjective optimization by providing a real-world benchmark test suite with complex variable interactions and correlations to different objectives. To the best of our knowledge, this is the first time to formulate a real-world problem into a benchmark test suite for large-scale multiobjective optimization, and it is also the first work proposing to solve TREE problems via evolutionary multiobjective optimization.

In the last two decades, a variety of different types of multiobjective optimization problems (MOPs) have been extensively investigated in the evolutionary computation community. However, most existing evolutionary algorithms encounter difficulties in dealing with MOPs whose Pareto optimal solutions are sparse (i.e., most decision variables of the optimal solutions are zero), especially when the number of decision variables is large. Such large-scale sparse MOPs exist in a wide range of applications, for example, feature selection that aims to find a small subset of features from a large number of candidate features, or structure optimization of neural networks whose connections are sparse to alleviate overfitting. This paper proposes an evolutionary algorithm for solving large-scale sparse MOPs. The proposed algorithm suggests a new population initialization strategy and genetic operators by taking the sparse nature of the Pareto optimal solutions into consideration, to ensure the sparsity of the generated solutions. Moreover, this paper also designs a test suite to assess the performance of the proposed algorithm for large-scale sparse MOPs. The experimental results on the proposed test suite and four application examples demonstrate the superiority of the proposed algorithm over seven existing algorithms in solving large-scale sparse MOPs.

In the last decade, the detection of overlapping communities has received increasing attention in network science. Among various clustering techniques, the fuzzy clustering has been widely adopted in overlapping community detection, since the soft assignment provided by it naturally meets the overlapping between multiple communities. The crucial step of fuzzy-clustering-based overlapping community detection is to find the optimal community centers, so that the overlapping communities can be obtained according to the membership degrees between nodes and community centers. In this article, we propose an evolutionary multiobjective optimization-based fuzzy method for overlapping community detection. In contrast to traditional fuzzy clustering methods, the proposed method optimizes the community centers by using a specially tailored multiobjective evolutionary algorithm. Moreover, it can also find an appropriate fuzzy threshold for each node, so that diverse overlapping community structures can be uncovered. In the experiments, we compare the proposed method with six state-of-the-art overlapping community detection approaches on synthetic and real-world networks with different scales and characteristics. The statistical results demonstrate that the proposed method can obtain the best results on most test instances.

The shelter location is very important for evacuation planning in natural disasters, and evolutionary algorithms (EAs) have demonstrated their effectiveness in solving this challenging problem. However, few EAs have been reported focusing on the shelter location problem under uncertainty of road networks due to the expensive cost of calculating the evacuation distance for individual evaluation. To address this issue, in this article, we propose a clustering-based surrogate-assisted multiobjective EA, termed AR-MOEA+SA, in the framework of a recently developed EA AR-MOEA. In AR-MOEA+SA, a surrogate model, the radial basis function (RBF), is adopted to approximately calculate the evacuation distance under uncertainty of road networks. Due to the fact that there often exist a large number of communities needing to be considered in shelter location, a clustering strategy is suggested to convert the surrogate of high-dimensional problem into the one of low-dimensional problem in the proposed AR-MOEA+SA for efficiently building the RBF network. A population initialization strategy is also suggested in AR-MOEA+SA to enhance the quality of training data in the early stages of evolution. Experimental results on a variety of test instances demonstrate the superiority of the proposed AR-MOEA+SA over the original version of AR-MOEA in terms of both computational efficiency and solution quality.

Many real-world optimization problems are characterized by multiple conflicting objectives, which are known as multi-objective optimization problems (MOPs). In the last two decades, evolutionary algorithms have shown promising performance in solving various MOPs, and a large number of multi-objective evolutionary algorithms (MOEAs) have been proposed. In order to determine the most suitable MOEA for a specific MOP, it is usually necessary to perform experiments to compare the performance of multiple candidate MOEAs. In 2017, an evolutionary multi-objective optimization platform was proposed by us, called PlatEMO, which provides the source codes of many state-of-the-art MOEAs and helps researchers perform batch experiments on these MOEAs. In this work, we illustrate the method of using the newest version of PlatEMO to solve MOPs in applications, by means of a case study on the feature selection problem, which is an important and difficult task in machine learning and data mining. This paper details the method of adding the feature selection problem to PlatEMO, and presents the experimental results of eight MOEAs on nine datasets in feature selection.

In this paper, we propose a framework to accelerate the computational efficiency of evolutionary algorithms on large-scale multiobjective optimization. The main idea is to track the Pareto optimal set (PS) directly via problem reformulation. To begin with, the algorithm obtains a set of reference directions in the decision space and associates them with a set of weight variables for locating the PS. Afterwards, the original large-scale multiobjective optimization problem is reformulated into a low-dimensional single-objective optimization problem. In the reformulated problem, the decision space is reconstructed by the weight variables and the objective space is reduced by an indicator function. Thanks to the low dimensionality of the weight variables and reduced objective space, a set of quasi-optimal solutions can be obtained efficiently. Finally, a multiobjective evolutionary algorithm is used to spread the quasi-optimal solutions over the approximate Pareto optimal front evenly. Experiments have been conducted on a variety of large-scale multiobjective problems with up to 5000 decision variables. Four different types of representative algorithms are embedded into the proposed framework and compared with their original versions, respectively. Furthermore, the proposed framework has been compared with two state-of-the-art algorithms for large-scale multiobjective optimization. The experimental results have demonstrated the significant improvement benefited from the framework in terms of its performance and computational efficiency in large-scale multiobjective optimization.

Surrogate-assisted evolutionary algorithms (SAEAs) have been developed mainly for solving expensive optimization problems where only a small number of real fitness evaluations are allowed. Most existing SAEAs are designed for solving low-dimensional single or multiobjective optimization problems, which are not well suited for many-objective optimization. This paper proposes a surrogate-assisted many-objective evolutionary algorithm that uses an artificial neural network to predict the dominance relationship between candidate solutions and reference solutions instead of approximating the objective values separately. The uncertainty information in prediction is taken into account together with the dominance relationship to select promising solutions to be evaluated using the real objective functions. Our simulation results demonstrate that the proposed algorithm outperforms the state-of-the-art evolutionary algorithms on a set of many-objective optimization test problems.

Both convergence and diversity are crucial to evolutionary many-objective optimization, whereas most existing dominance relations show poor performance in balancing them, thus easily leading to a set of solutions concentrating on a small region of the Pareto fronts. In this paper, a novel dominance relation is proposed to better balance convergence and diversity for evolutionary many-objective optimization. In the proposed dominance relation, an adaptive niching technique is developed based on the angles between the candidate solutions, where only the best converged candidate solution is identified to be nondominated in each niche. Experimental results demonstrate that the proposed dominance relation outperforms existing dominance relations in balancing convergence and diversity. A modified NSGA-II is suggested based on the proposed dominance relation, which shows competitiveness against the state-of-the-art algorithms in solving many-objective optimization problems. The effectiveness of the proposed dominance relation is also verified on several other existing multi- and many-objective evolutionary algorithms.

The effectiveness of evolutionary algorithms have been verified on multi-objective optimization, and a large number of multi-objective evolutionary algorithms have been proposed during the last two decades. To quantitatively compare the performance of different algorithms, a set of uniformly distributed reference points sampled on the Pareto fronts of benchmark problems are needed in the calculation of many performance metrics. However, not much work has been done to investigate the method for sampling reference points on Pareto fronts, even though it is not an easy task for many Pareto fronts with irregular shapes. More recently, an evolutionary multi-objective optimization platform was proposed by us, called PlatEMO, which can automatically generate reference points on each Pareto front and use them to calculate the performance metric values. In this paper, we report the reference point sampling methods used in PlatEMO for different types of Pareto fronts. Experimental results show that the reference points generated by the proposed sampling methods can evaluate the performance of algorithms more accurately than randomly sampled reference points.

During the past two decades, a variety of multiobjective evolutionary algorithms (MOEAs) have been proposed in the literature. As pointed out in some recent studies, however, the performance of an MOEA can strongly depend on the Pareto front shape of the problem to be solved, whereas most existing MOEAs show poor versatility on problems with different shapes of Pareto fronts. To address this issue, we propose an MOEA based on an enhanced inverted generational distance indicator, in which an adaptation method is suggested to adjust a set of reference points based on the indicator contributions of candidate solutions in an external archive. Our experimental results demonstrate that the proposed algorithm is versatile for solving problems with various types of Pareto fronts, outperforming several state-of-the-art evolutionary algorithms for multiobjective and many-objective optimization.

The current literature of evolutionary many-objective optimization is merely focused on the scalability to the number of objectives, while little work has considered the scalability to the number of decision variables. Nevertheless, many real-world problems can involve both many objectives and large-scale decision variables. To tackle such large-scale many-objective optimization problems (MaOPs), this paper proposes a specially tailored evolutionary algorithm based on a decision variable clustering method. To begin with, the decision variable clustering method divides the decision variables into two types: 1) convergence-related variables and 2) diversity-related variables. Afterward, to optimize the two types of decision variables, a convergence optimization strategy and a diversity optimization strategy are adopted. In addition, a fast nondominated sorting approach is developed to further improve the computational efficiency of the proposed algorithm. To assess the performance of the proposed algorithm, empirical experiments have been conducted on a variety of large-scale MaOPs with up to ten objectives and 5000 decision variables. Our experimental results demonstrate that the proposed algorithm has significant advantages over several state-of-the-art evolutionary algorithms in terms of the scalability to decision variables on MaOPs.

In this paper, we propose a sampling approach of reference points used for performance metrics of multi-objective evolutionary algorithms. Traditional reference point sampling methods, such as the Das and Dennis method, usually sample the reference points via a set of uniformly distributed weight vectors generated on an ideal hyper-plane in objective space, which however often ignore the geometric shape of a specific Pareto front. Therefore, we propose a novel reference point sampling approach by taking the specific shape of the Pareto optimal front to be tackled into account for measuring the performance of multi-objective evolutionary algorithms. The performance of the proposed reference point sampling method against the other two state-of-the-art sampling methods is tested on six test instances in various conditions, which clearly demonstrate the effectiveness and superiority of the proposed sampling method.

As a pivotal component in multi-objective evolutionary algorithms (MOEAs), the environmental selection determines the quality of candidate solutions to survive at each generation. In practice, different environmental selection strategies can be based on different selection criteria, where the performance metrics (or indicators) are shown to be among the most effective ones. This paper proposes an MOEA whose environmental selection is based on an enhanced inverted generational distance metric that is able to detect noncontributing solutions (termed IGD-NS), thereby considerably accelerating the convergence of the evolutionary search. Experimental results on ZDT and DTLZ test suites demonstrate the competitive performance of the proposed MOEA/IGD-NS in comparison with some representative MOEAs.

Evolutionary algorithms have been shown to be powerful for solving multiobjective optimization problems, in which nondominated sorting is a widely adopted technique in selection. This technique, however, can be computationally expensive, especially when the number of individuals in the population becomes large. This is mainly because in most existing nondominated sorting algorithms, a solution needs to be compared with all other solutions before it can be assigned to a front. In this paper we propose a novel, computationally efficient approach to nondominated sorting, termed efficient nondominated sort (ENS). In ENS, a solution to be assigned to a front needs to be compared only with those that have already been assigned to a front, thereby avoiding many unnecessary dominance comparisons. Based on this new approach, two nondominated sorting algorithms have been suggested. Both theoretical analysis and empirical results show that the ENS-based sorting algorithms are computationally more efficient than the state-of-the-art nondominated sorting methods.

Evolutionary algorithms (EAs) have shown to be promising in solving many-objective optimization problems (MaOPs), where the performance of these algorithms heavily depends on whether solutions that can accelerate convergence toward the Pareto front and maintaining a high degree of diversity will be selected from a set of nondominated solutions. In this paper, we propose a knee point-driven EA to solve MaOPs. Our basic idea is that knee points are naturally most preferred among nondominated solutions if no explicit user preferences are given. A bias toward the knee points in the nondominated solutions in the current population is shown to be an approximation of a bias toward a large hypervolume, thereby enhancing the convergence performance in many-objective optimization. In addition, as at most one solution will be identified as a knee point inside the neighborhood of each solution in the nondominated front, no additional diversity maintenance mechanisms need to be introduced in the proposed algorithm, considerably reducing the computational complexity compared to many existing multiobjective EAs for many-objective optimization. Experimental results on 16 test problems demonstrate the competitiveness of the proposed algorithm in terms of both solution quality and computational efficiency.