Engineering optimization problems are often nonlinear, high-dimensional, and constrained, making them challenging for conventional optimization techniques. Although L-SHADE, an adaptive differential evolution (DE) algorithm with success-history based parameter adaptation, has demonstrated competitive performance, it still suffer from limited population diversity and weak local exploitation, leading to an imbalance between exploration and exploitation in complex optimization. To address these limitations, this paper proposes LCO-LSHADE-GSRL, a novel DE variant that enhances both global exploration and local exploitation capabilities. The proposed algorithm integrates three key components: (1) a Logistic Chaos Orthogonal Initialization mechanism that improves initial population diversity and ensures uniform coverage of the search space. (2) a GAN-driven Specular Reflection Learning (SRL) mechanism that effectively escapes from local optima. (3) a design that adapts effectively to constrained optimization scenarios. Comprehensive experiments conducted on the CEC 2019 and 2022 benchmark suites demonstrate that LCO-LSHADE-GSRL exhibits superior convergence performance, solution accuracy, and robustness compared to L-SHADE, LSHADE-cnEpSin, and WOA, GJO, PO, PIMO, and CDO. Furthermore, in three real-world engineering problems–speed reducer, step-cone pulley, and hydrostatic thrust bearing, which reduces system weight and power loss while satisfying all design constraints. These results demonstrate its potential for solving complex engineering optimization tasks with high reliability and efficiency.

Neural Architecture Search (NAS) has emerged as an important area in deep learning since it can automatically design high performance network architectures, where Evolution-based NAS (EvoNAS) has made great progress due to the efficient optimization ability of evolutionary algorithms. However, EvoNAS requires evaluating the architectures formed by individuals in the population, and it is inevitable to consume a large amount of evaluation time and computational resources, resulting in restricting the applicability of EvoNAS. To solve the above problems, this paper proposes a Weight Inheritance and Guided Strategy based Evolutionary Network Architecture Search (WIGEvoNAS). Firstly, based on existing manually designed networks, an expanded search space is designed, which includes new convolution operations. Secondly, a weight inheritance strategy is proposed to reduce the training time of candidate architectures in each generation. Finally, a guidance mutation strategy is proposed to direct population evolution towards architectures with superior for the purpose of generating better offspring. The proposed method is compared with several state-of-the-art NAS methods and manual networks on the CIFAR-10, CIFAR-100 and the NASBench-201 benchmark datasets. The empirical results demonstrate that the proposed method achieves promising performance, with error rates of 2.52% on CIFAR-10 and 15.43% on CIFAR-100 respectively. Moreover, the proposed method significantly reduces search costs to 0.9 GPU-days.

Dynamic constrained multiobjective optimization problems (DCMOPs) are widely existed in real-world applications and emerged as a prominent research focus in the evolutionary computation community. Current studies on DCMOPs face two main challenges: limited accuracy in population prediction, and a lack of effective strategies to improve static optimizer performance. To tackle these challenges, this paper proposes a dynamic constrained multiobjective evolutionary algorithm based on multipopulation prediction and dynamic fusion ranking. Specifically, an efficient computer-vision-inspired point set registration method, named coherent point drift, is introduced to align individuals across successive environments. With the correspondences between two environments, the solution trajectory tracking problem is transformed as a point set registration problem. Based on the observed trajectory of solutions, the Pareto-optimal set or Pareto-optimal front in the new environment can be predicted. Additionally, this paper highlights the importance of task-specific multipopulation prediction. After analysis the specific tasks of each population, different initial populations tailored to the tasks are predicted by the proposed prediction method. Finally, a dynamic fusion based two-ranking environmental selection strategy is proposed for the auxiliary task. This strategy dynamically integrates experience-based and constraint-based approaches, improving the auxiliary population evolutionary efficiency and its alignment with the main task. The superiority of proposed algorithm is validated through extensive experiments on a series of benchmark problems and a real-world raw ore allocation problem.

In practical scenarios, there may be solutions in the decision space with close objective values but located far apart, a characteristic known as multimodal multiobjective problems (MMOPs). While most multimodal multiobjective evolutionary algorithms (MMEAs) focus on finding global Pareto optimal solution sets (PSs) and local PSs demonstrating satisfactory convergence performance, decision-makers in real-world scenarios are often also interested in local PSs that exhibit strong robustness. In this study, we propose several benchmark functions in which the global and local PSs have varying levels of robustness. Then, we introduce an innovative dual-population evolutionary algorithm, termed GLR-MMEA, designed to simultaneously find both global PSs and local PSs with strong robustness. In GLR-MMEA, the convergence population focuses on identifying global PSs, providing convergence information to the diversity population. Meanwhile, the diversity population manages the detection of both global PSs and local PSs with strong robustness. In the process of updating the diversity population, a robustness indicator is proposed to access the robustness of solutions. Furthermore, a selection mechanism founded on this robustness indicator is applied to identify local PSs with high robustness. The experimental results show that GLR-MMEA performs competitively against other leading MMEAs in working on the selected benchmark functions.

In multiobjective feature selection, different feature subsets with the same number of selected features can achieve identical classification accuracy, meaning that it is a multimodal optimization problem. To effectively search for multimodal feature subsets within the vast search spaces of high-dimensional datasets, it is crucial to adopt reasonable encoding and search methods. Generally, applying a uniform evolutionary operator based on a single encoding method across the entire feature space is inefficient and prone to falling into local optima. To address the above issues, this article proposes a multimodal multiobjective feature selection method based on a joint encoding mechanism that combines discrete encoding and continuous encoding. It provides new perspectives to solve the high-dimensional feature selection problem from encoding methods to search operators. First, the search space is divided into a discrete encoding region and a continuous encoding region based on the knee points of feature importance ranking curve. A tailored initialization strategy is used to obtain the initial population for joint encoding. Second, an adaptive niche strategy based on three priorities is proposed, which ensures the similarity of individuals within a niche and the difference between niches. In addition, different search operators are cooperated with the two encoding strategies, respectively, to achieve effective and efficient search. The experimental results on 24 datasets show that the proposed algorithm achieves a better-classification performance than the state-of-the-art feature selection methods.

Dynamic constrained multiobjective optimization problems (DCMOPs) are widely encountered in real-world applications and have attracted increasing attention in the evolutionary computation community. Existing studies primarily focus on the population initialization, but disregard to improve the evolution process. In DCMOPs, the knowledge is undoubtedly more abundant as the existence of historical environments. Therefore, extracting and utilizing useful knowledge from historical environments can further improve the evolution process. In this article, a history-assisted two-state auxiliary task collaboration approach is proposed to solve DCMOPs by conducting a more effective auxiliary task. Specifically, the algorithm indicates that constrained Pareto-optimal front (CPF) in similar historical environments is more suitable as an auxiliary task for current evolution, as it is closer to the current CPF. To identify the most suitable environment from extensive historical environments, a novel prediction-based similar environment identification method is proposed for the auxiliary task. To fully utilize the novel auxiliary task, an experience-driven two-state environmental selection strategy is proposed, which conditionally considers both historical and current information. In this strategy, individuals in state A are inclined to promote toward the historical CPF while disregarding the dominance relationship or constraints, to help cross infeasible region and approach CPF rapidly. For those individuals in state B, dominance relationship is taken into consideration, further bringing auxiliary task closer to the main task. The superiority of the proposed algorithm has been comprehensively demonstrated by experimental comparison results on a variety of benchmark test problems and a real-world problem with raw ore allocation in mineral processing.

Road extraction from remote sensing images is crucial for various applications, such as urban planning and traffic management. Exiting road extraction methods face limitations, such as the need to improve accuracy or high requirements for training data. To address these limitations, this paper proposes a two-stage genetic programming (GP) approach to feature construction and ensemble evolving for road extraction from remote sensing images. The proposed approach can automatically evolve interpretable solutions with small training data and achieve end-to-end road extraction from remote sensing images. In the first stage, a feature construction strategy is designed to construct high-level features from multiple views, respectively, based on the type of features. In the second stage, a novel GP individual representation is proposed to allow the new method to combine the high-level features of different types, automatically generate an ensemble of base classification methods and select their parameters for classification. The proposed approach is examined on eleven road extraction tasks from the Deepglobe dataset, and compared with seven traditional methods, nine GP-based methods, and four neural network methods. The experimental results demonstrate the effectiveness of the proposed approach in enhancing accuracy and generalization capabilities. Furthermore, the visualization analysis shows the good interpretability of solution/model of the proposed approach.

Unbalanced classification problems have attracted significant academic attention due to their widespread existence in the real world. The lack of recognition accuracy of minority class samples and the “curse of dimensionality” are two major difficulties in unbalanced classification problems. Existing unbalanced classification methods run the risk of losing the original feature information and are prone to bias toward the majority class. Multiform optimization is famous for capturing useful knowledge from alternative forms to help solve the original task. Motivated by this, this article introduces a multiform evolutionary framework that addresses the issue of multiobjective feature selection in unbalanced classification scenarios. It aims to utilize the advanced experience of selecting features on balanced datasets to assist in the search for feature subsets that can more accurately identify minority classes on the original dataset. Specifically, a knowledge transfer strategy is proposed to draw on the search experience of the auxiliary task from the oversampled dataset to help the cost-sensitive learning task based on the original dataset jump out of the local optimum. In addition, an offspring repairing mechanism is proposed to filter redundant features by considering the frequency of selected features. Experimental results on 23 real-world benchmark datasets demonstrate that the proposed method can select fewer features and achieve better classification results compared to six state-of-the-art multiobjective feature selection algorithms and three classical oversampling algorithms. Furthermore, the difference in performance of four base classifiers is investigated through a series of comparative experiments.

The censored regression (CR)-based normalized subband adaptive algorithm (CR-NSAF) model has been recently introduced for processing signals with censored data. However, the effectiveness of this algorithm declines when dealing with noisy input signals in impulsive noise environments. To resolve this challenge, we propose a robust bias-compensated CR-NSAF algorithm (RBC-CRNSAF). This algorithm alleviates the negative impacts of the CR system and improves robustness by employing a logarithmic cost function approach. It also minimizes estimation bias from input noise by incorporating new compensation terms into the weights update function. Additionally, we analyze the computational complexity, convergence characteristics, and stability conditions of the algorithm. Finally, computer simulations indicate that RBC-CRNSAF considerably outperforms other similar algorithms in impulsive noise environments, validating its enhanced performance.

Feature selection in machine learning as a key data processing technique has two conflicting goals: minimizing the classification error rate and minimizing the number of features selected. However, most of the existing multi-objective feature selection methods face the problems of easily falling into local optima and slow convergence by virtue of their problem characteristics, such as partially conflicting objectives and highly discontinuous Pareto fronts. To solve these problems, this article proposes a multiform optimization framework to solve a multi-objective feature selection task together with several auxiliary single-objective feature selection tasks in a multitask environment. The proposed framework uses the problem-solving experience of single-objective tasks to assist the multi-objective feature selection task in exploring more promising regions and accelerating the convergence speed. Specifically, a knowledge transfer strategy based on the search experience of different tasks is developed to accomplish multiform optimization. In addition, a diversity enhancement mechanism is presented to improve search ability in promising decision space areas by considering historical information about the population. In most cases, the experiment results on 27 datasets demonstrate that the proposed technique can uncover more diversified feature subsets on the Pareto front in less time than existing state-of-the-art methods.

Evolutionary constrained multiobjective optimization has received extensive attention and research in the past two decades, and a lot of benchmarks have been proposed to test the effect of the constrained multiobjective evolutionary algorithms (CMOEAs). Specially, the constraint functions are highly correlated with the objective values, which makes the features of constraints too monotonic and differ from the properties of the real-world problems. Accordingly, previous CMOEAs cannot solve real-world problems well, which generally involve decision space constraints with multi-modal/non-linear features. Therefore, we propose a new benchmark framework and design a suite of new test functions with scalable high-dimensional decision space constraints. To be specific, different high-dimensional constraint functions and mixed linkages in variables are considered to be close to realistic features. In this framework, several parameter interfaces are provided, so that users can easily adjust the parameters to obtain the variant functions and test the generalization performance of the algorithms. Different types of existing CMOEAs are employed to test the use of the proposed test functions, and the results show that they are easy to fall into local feasible regions. Therefore, we improve one evolutionary multitasking-based CMOEA to better handle these problems, in which a new search algorithm is designed to enhance the search abilities of populations. Compared with the existing CMOEAs, the proposed CMOEA presents better performance.

Dynamic multiobjective optimization problems (DMOPs) are widely encountered in real-world applications and have received considerable attention in recent years. During the process of solving DMOPs, tracking the constantly changing Pareto optimal set (POS) quickly is the main task, and numerous methods have been proposed to achieve this task from different perspectives. However, how to improve the evolution of static optimizers via using historical information from past environments gets little attention. In fact, with adequate historical evolution information to learn, the trend of population evolution in the historical environments is promising to guide the future evolution and thus enhance the search ability. Therefore, in this paper, a historical evolution learning based framework is proposed to assist the static optimizers in fully using the historical evolution direction and POSs distribution. Specifically, two new models are designed with a purpose of generating offsprings and environmental selection, respectively. For enhancing the contribution of offsprings, a direction guidance model is developed to guide the offsprings according to the direction tendency of historical evolution. Furthermore, to improve the robustness of static optimizers and avoid the disturbance caused by misleading solutions, a manifold revise model is proposed to produce promising solutions via consulting historical POSs distribution. The proposed framework employs these two models collaboratively, and it is flexible and readily to be embedded into various dynamic response mechanisms and static optimizers in coping with DMOPs. The superiority of the framework has been comprehensively demonstrated by experimental comparison results on a variety of benchmark test problems.

Feature construction has shown promise in improving the accuracy of crop classification by constructing high-level features. However, current feature construction methods often rely on domain knowledge and have a limited interpretability of the solutions. To address this, this study proposes a new genetic programming (GP) approach to automatically evolve solutions with high interpretability that can construct high-level features for crop classification from hyperspectral images. A flexible representation of multiple trees is proposed in the proposed GP approach to construct various types of high-level features from the original ones, simultaneously. To improve the search ability, a new offspring generation method is developed to dynamically guide the evolution of the population while improving the diversity of the population. The new approach wraps with three classification algorithms, i.e., support vector machine (SVM), naive Bayes (NB), and k-nearest neighbor (KNN), for crop classification on three datasets with different difficulties and tasks. The results demonstrate that the features constructed by the new approach can effectively distinguish different crop categories. The new approach achieves better performance than the compared GP-based method, classic methods, and deep learning methods in crop classification using hyperspectral images. Importantly, the proposed approach shows the high interpretability of the constructed features.

Constrained multiobjective optimization problems with deceptive constraints (DCMOPs) are a kind of complex optimization problems and have received some attention. For DCMOPs, the closer a solution is to the feasible region, the larger its constraint value. Moreover, multiple local infeasible regions will have different minimal constraint values according to their distances to feasible regions. Therefore, most of the existing algorithms are easy to fall into local regions, and even cannot find any feasible solution. To address DCMOPs, this article proposes a new evolutionary multitasking algorithm with a cooperative multistep mutation strategy. In this algorithm, the DCMOP is transformed into a multitasking optimization problem, in which the main task is the original DCMOP and the created auxiliary task aims to provide effective help for solving the main task. Specially, the designed cooperative multistep mutation strategy contains two contributions to solve deceptive constraints. First, a multistep mechanism is proposed, in which the individuals will use multiple different steps to generate the multiple offspring solutions along one direction, so as to expand search range to find feasible regions. Second, a cooperative mechanism between the two tasks is proposed, in which the main purpose is to provide effective and stable search directions. To be specific, an opposite solution generation method is utilized to generate the opposite solution of auxiliary population in the search space, and the direction from the auxiliary population to the main population will be formed. Combined with these two mechanisms, the proposed cooperative multistep mutation strategy can effectively improve the population diversity along the promising and stable search directions. In the experiments, the proposed algorithm is tested on the two benchmark DCMOPs, which contain objective space constraints and decision space constraints respectively. The results show the effectiveness and superiority of the proposed algorithm over the latest compared algorithms.

Improving classification performance and reducing the number of selected features are two conflicting objectives of feature selection, which can be well solved by multiobjective algorithms. However, as the dimensionality of the data increases, the search space for feature selection will grow exponentially, which leads to high-computational costs. Additionally, the complex interaction among features makes the population prone to falling into local optimal. To address these issues, feature grouping can treat one dimension as a group of features instead of one feature, effectively transforming the high-dimensional search space into a lower-dimensional one. Since different grouping forms can be converted into different feature combination spaces, the search directions of the population also vary. Inspired by this, a multiform optimization approach based on space transformation (MOFS-MST) is proposed in this article. Specifically, two different grouping forms are set based on the ranking of features in different evaluation criteria to construct a multiform framework, thereby increasing the diversity of the population. During the evolutionary process, a knowledge transfer strategy based on feature groups is executed between the two forms of grouping in order to help each other escape local optima. Moreover, it can dynamically adjust the state of feature grouping to enhance the potential for feature interaction. Experimental results demonstrate that this method outperforms six other state-of-the-art multiobjective high-dimensional feature selection methods on 12 high-dimensional datasets.

Financial technology (FinTech) has become a hot research topic recently, moreover, an area of major investment for most financial institutions. As the FinTech unleashed many new strategic solutions for major financial problems. One of those pivotal strategies of FinTech is financial crisis prediction (FCP) that dictates the financial status of an institution. Also, the rise of the Internet-of-Things (IoT) technology has paved a new way for interaction between humans and the physical world. Therefore, IoT can feasibly be incorporated into the FCP model to obtain a real-time analysis of the financial data from the clients. With this perspective, we propose an intelligent IoT-aided FCP model using metaheuristic algorithms. The proposed FCP method comprises data acquisition, preprocessing, feature selection (FS), and classification. First, the financial data of the enterprises are collected using IoT devices, such as smartphones, laptops, etc. Next, the quantum artificial butterfly optimization (QABO) approach for FS is applied to choose an optimal set of features. Afterward, long short-time memory (LSTM) with recurrent neural network (RNN) model is employed to classify the collected financial data. An exhaustive experimental validation process is carried out to ensure the performance of the proposed QABO-LSTM-RNN model. The simulation results accredited the efficacy of the proposed model contrasted with other baseline methods in terms of best cost, sensitivity, specificity, accuracy, F-score, kappa, and Mathew correlation coefficient (MCC).

As a key preprocessing step in classification, feature selection involves two conflicting objectives: maximizing the classification accuracy and minimizing the number of selected features. Therefore, multi-objective optimization is widely used in feature selection due to its excellent trade-off between the convergence of two objectives. However, most existing multi-objective feature selection methods still face the issues of the “curse of dimensionality” and high computational costs, especially when the search space is large. To solve the above issues, this paper proposes a bidirectional dynamic grouping multi-objective evolutionary approach for high-dimensional feature selection, referred to as BDGMOEA. This approach transforms a high-dimensional feature selection problem into a feature selection task with a smaller search space by the idea of feature grouping, in which one bit of an individual represents a group of features. Specifically, a grouping search strategy is developed to divide the features into different quadrants according to the importance of the features obtained by different evaluation techniques. Then, the features in each quadrant are grouped by sector. This strategy can effectively narrow the search space and quickly locate promising feature regions. In addition, a bidirectional dynamic adjustment mechanism is presented by considering the evolutionary state of the population, and it can be used to explore each feature in more detail and comprehensively to prevent good features from being ignored in unselected groups. The experimental results demonstrate that the proposed BDGMOEA method performs the best in most cases, indicating that BDGMOEA not only achieves better classification performance but also reduces the training time.

Constrained multiobjective optimization problems (CMOPs) involve multiple objectives to be optimized and various constraints to be satisfied, which challenges the evolutionary algorithms in balancing the objectives and constraints. This article attempts to explore and utilize the relationship between constrained Pareto front (CPF) and unconstrained Pareto front (UPF) to solve CMOPs. Especially, for a given CMOP, the evolutionary process is divided into the learning stage and the evolving stage. The purpose of the learning stage is to measure the relationship between CPF and UPF. To this end, we first create two populations and evolve them by specific learning strategies to approach the CPF and UPF, respectively. Then, the feasibility information and dominance relationship of the two populations are used to determine the relationship. Based on the learned relationship, specific evolving strategies are designed in the evolving stage to improve the utilization efficiency of objective information, so as to better solve this CMOP. By the above process, a new constrained multiobjective evolutionary algorithm (CMOEA) is presented. Comprehensive experimental results on 65 benchmark functions and ten real-world CMOPs show that the proposed method has a better or very competitive performance in comparison with several state-of-the-art CMOEAs. Moreover, this article demonstrates that using the relationship between CPF and UPF to guide the utilization of objective information is promising in solving CMOPs.

Handling constrained multiobjective optimization problems (CMOPs) is extremely challenging, since multiple conflicting objectives subject to various constraints require to be simultaneously optimized. To deal with CMOPs, numerous constrained multiobjective evolutionary algorithms (CMOEAs) have been proposed in recent years, and they have achieved promising performance. However, there has been few literature on the systematic review of the related studies currently. This article provides a comprehensive survey for evolutionary constrained multiobjective optimization. We first review a large number of CMOEAs through categorization and analyze their advantages and drawbacks in each category. Then, we summarize the benchmark test problems and investigate the performance of different constraint handling techniques (CHTs) and different algorithms, followed by some emerging and representative applications of CMOEAs. Finally, we discuss some new challenges and point out some directions of the future research in the field of evolutionary constrained multiobjective optimization.

When solving constrained multiobjective optimization problems (CMOPs), the utilization of infeasible solutions significantly affects algorithm’s performance because they not only maintain diversity but also provide promising search directions. In light of this situation, this article proposes a new multitasking-constrained multiobjective optimization (MTCMO) framework, in which a dynamic auxiliary task is created to assist in solving a complex CMOP (the main task) via the knowledge transfer. Moreover, the constraint boundary of the auxiliary task reduces dynamically, so that it keeps a high relatedness with the main task to continuously provide supplementary evolutionary directions. Furthermore, an improved$\epsilon $method is designed for the auxiliary task to utilize diverse high-quality infeasible solutions for breaking through infeasible obstacles in the early stage and approaching the feasible boundary from infeasible regions in the later stage. Besides, a new test function with decision space constraints is designed, where one parameter can be adjusted to control the overlap degree between the constrained Pareto front and the unconstrained Pareto front. This function and the other two modified existing functions are used to analyze the characteristics of MTCMO. Finally, compared with 11 state-of-the-art peer methods, the superior or competitive performance of MTCMO is demonstrated on 54 benchmark functions and two real-world applications.

The evolutionary algorithm recommendation is catching increasing attention when solving practical application problems since different algorithms often perform differently on different problems. To achieve the algorithm recommendation, extracting effective features to accurately characterize the problems is necessary, which is related to the feature extraction problem. So far, most feature extraction methods focus on single-objective optimization problems, and only a few studies are conducted on multiobjective optimization problems and constrained optimization problems, let alone constrained multiobjective optimization problems (CMOPs) that are widely encountered in the real world. To fill the gap, this article proposes an evolution-based constrained multiobjective feature extraction method (ECMOFE), in which the information generated in the evolutionary process is leveraged to form the feature matrix. To be specific, we create two populations to, respectively, optimize constraints and objectives for some generations. Furthermore, two complementary evolutionary operators are used to generate offspring for each population. In the environmental selection, the successful rate of offspring individuals generated by each operator of each population is recorded to form the feature matrix. Then, a dimension reduction method is designed to compress the size of the feature matrix. By the above process, the feature vector that can reflect the global relationship between constraints and objectives and the difficulty of the CMOP is formed. Based on the formed features, several algorithm recommendation methods are built on the basis of classifiers. The results based on multiple metrics show the effectiveness of the proposed ECMOFE.

Dynamic multiobjective optimization problems (DMOPs) are characterized by their multiple objectives, constraints, and parameters that may change over time. The challenge in solving DMOPs is how to track the varying Pareto optimal solution sets quickly and accurately. Therefore, an inter-individual correlation and dimension-based dual learning method is proposed in this paper. Two learning strategies, decomposition-based inter-individual correlation transfer learning (DICTL) and dimension-wise learning (DL), are developed to respectively generate one-half of the initial population in the new environment. More specifically, DICTL learns the inter-individual correlation from the final population of the adjacent environment and then transfers it to the new environment, aiming to maintain the diversity and distribution of the predicted population. While DL extracts the changing pattern of dynamic environments from the high-quality solutions of historical environments in the perspective of variable dimension, trying to improve the quality of the population and accelerate the convergence. The designed two learning strategies (DICTL&DL) work complementarily and collaboratively to make the algorithm adapt to dynamic environments better and faster. Comprehensive experiments have been conducted by comparing the proposed method with four state-of-the-art algorithms on 14 benchmark problems. The results demonstrate the superiority of the proposed method.

When performing feature selection on most data sets, there is a general situation that some different feature subsets have the same number of selected features and classification error rate. This indicates that feature selection in some data sets is a multimodal multiobjective optimization (MMO) problem. Most of the current studies on feature selection ignore the MMO problems. Therefore, this paper proposes a feature selection method based on a multimodal multiobjective genetic algorithm (MMOGA) to solve the problem. This algorithm is mainly improved in three aspects. First, a special initialization strategy based on symmetric uncertainty is designed to improve the fitness of the initial population. Second, this paper adds a niche strategy to the genetic algorithm to search for multimodal solutions. Unlike traditional niche methods that has a central individual, this algorithm also considers the distances between individuals in the niche. Third, to effectively utilize excellent individuals for evolution, this algorithm uses a method based on the Pareto set of the niche to generate offspring. Finally, by comparing with other algorithms, the effectiveness of the MMOGA in feature selection is verified. This algorithm can successfully find equivalent feature subsets on different datasets.

When addressing constrained multiobjective optimization problems (CMOPs) via evolutionary algorithms, various constraints and multiple objectives need to be satisfied and optimized simultaneously, which causes difficulties for the solver. In this article, an evolutionary multitasking (EMT)-based constrained multiobjective optimization (EMCMO) framework is developed to solve CMOPs. In EMCMO, the optimization of a CMOP is transformed into two related tasks: one task is for the original CMOP, and the other task is only for the objectives by ignoring all constraints. The main purpose of the second task is to continuously provide useful knowledge of objectives to the first task, thus facilitating solving the CMOP. Specially, the genes carried by parent individuals or offspring individuals are dynamically regarded as useful knowledge due to the different complementarities of the two tasks. Moreover, the useful knowledge is found by the designed tentative method and transferred to improve the performance of the two tasks. To the best of our knowledge, this is the first attempt to use EMT to solve CMOPs. To verify the performance of EMCMO, an instance of EMCMO is obtained by employing a genetic algorithm as the optimizer. Comprehensive experiments are conducted on four benchmark test suites to verify the effectiveness of knowledge transfer. Furthermore, compared with other state-of-the-art constrained multiobjective optimization algorithms, EMCMO can produce better or at least comparable performance.

Solving constrained multiobjective optimization problems brings great challenges to an evolutionary algorithm, since it simultaneously requires the optimization among several conflicting objective functions and the satisfaction of various constraints. Hence, how to adjust the tradeoff between objective functions and constraints is crucial. In this article, we propose a dynamic selection preference-assisted constrained multiobjective differential evolutionary (DE) algorithm. In our approach, the selection preference of each individual is suitably switching from the objective functions to constraints as the evolutionary process. To be specific, the information of objective function, without considering any constraints, is extracted based on Pareto dominance to maintain the convergence and diversity by exploring the feasible and infeasible regions; while the information of constraint is used based on constrained dominance principle to promote the feasibility. Then, the tradeoff in these two kinds of information is adjusted dynamically, by emphasizing the utilization of objective functions at the early stage and focusing on constraints at the latter stage. Furthermore, to generate the promising offspring, two DE operators with distinct characteristics are selected as components of the search algorithm. Experiments on four test suites including 56 benchmark problems indicate that the proposed method exhibits superior or at least competitive performance, in comparison with other well-established methods.

Improving the accuracy of wind speed forecasting is essential for the usage of wind energy. This paper proposes an evolutionary ensemble learning (EEL) method, which consists of ensemble empirical mode decomposition (EEMD), random vector functional link network (RVFL) based ensemble learning, and grid-based multimodal multi-objective evolutionary algorithm (MMOG). Based on MMOG, the proposed ensemble learning model is improved in terms of accuracy. Several benchmark forecast methods are compared with the proposed EEL model on 12 wind speed forecasting datasets. The experiment results validate the superiority of the proposed EEL model in wind speed forecasting.

Optical cloud radio access network (O-CRAN) is an emerging solution for IIoT, where numerous different devices/nodes are networked together. O-CRAN provides pool of shareable computing facility, equipped with hundreds of general-purpose processor (GPP). The GPPs process massive big data exerted by nodes via remote radio heads (RRHs), regarded as RRH-requests, which are bandwidth-intensive and deadline-constrained digitized base-band signals. Computing resource (CR) optimization has been widely investigated in O-CRAN. However, the existing optimizations may not guarantee workload and thermal balance among the active GPPs while satisfying RRH-request's deadline, which are necessary to efficiently leverage virtualization GPP capacity in a manner that provides the greatest uniform CR utilization (CRU). Due to varying network-load a single optimal solution does not exist. Therefore, in this article, we propose a modified-first-fit decreasing (MFFD) algorithm to obtain a suboptimal solution for each time_stage. The MFFD evenly assigns RRH-requests among GPPs that maximizes individual CRU uniformly contrasting with FFD.

Multiobjective optimization problems with multiple equivalent global Pareto solutions or with at least one local Pareto solution are called multimodal multiobjective optimization problems (MMOP). Most of the existing multimodal multiobjective algorithms can only find global Pareto solutions. However, the local Pareto solutions are of great significance when the global ones are impracticable. This paper proposes a Multimodal Multiobjective Genetic Algorithm (MMOGA) to find both global and local Pareto solutions. In MMOGA, only individuals in the same niche can mate and compete with each other, thus enabling the population to evolve in local areas. Experimental results show that the proposed algorithm can find both global and local Pareto sets of MMOPs.

A modified multi-objective harmony search algorithm called Niching Multi-objective Harmony Search Algorithm (NMOHSA) is proposed to solve multimodal multi-objective optimization problems. It adopts the neighborhood information to build dynamic harmony memory for maintaining the population diversity. A new memory consideration rule is also applied to prevent the algorithm be trapped into local optimal solution. Moreover, two key parameters, harmony memory consideration rate (HMCR) and pitch adjustment rate (PAR), are dynamically adjusted. Empirical results show that the proposed algorithm performs much better than the other existing multimodal multi-objective algorithms in terms of the solution quality.

A modified multi-objective bat algorithm called Performance Enhanced Niching Multi-objective Bat algorithm (PEN-MOBA) is proposed to solve multimodal multi-objective optimization problems. It adopts a dynamic ring topology to form stable niches for maintaining the population diversity, and integrates the stagnation detection strategy to improve the searching ability. The algorithm is compared with a number of state-of-the-art multimodal multi-objective optimizers on twelve multimodal multi-objective test functions. The experimental results verify that the proposed algorithm is effective multimodal multi-objective optimizers and outperforms the existing algorithms on the test functions.

Parameters identification of solar photovoltaic (PV) models, as a complex nonlinear optimization problem, has received more and more attention of many scholars. Although there have been already numerous techniques for this problem, it is still challenging to identify the model parameters accurately. For the purpose of improving the results of parameters identification of different photovoltaic models, a modified particle swarm optimization (MPSO) algorithm is proposed in this paper. In MPSO, in order to explore more promising regions of the search space, a mutation operation inspired by differential evolution is employed to improve the quality of personal best of each particle as well as the global best of the current population. Moreover, the damping bound-handling method is used to alleviate the premature convergence. The effectiveness of MPSO is validated via estimating parameters of the single diode, double diode, and photovoltaic module model, respectively. The simulation and experimental results comprehensively demonstrate the superiority of MPSO compared to other stateof-the-art algorithms.

In feature selection, the number of selected features and the classification accuracy are two common objectives to be optimized. However, few studies pay attention to which features are selected. In many feature selection problems, different feature subsets with the same number of selected features can achieve similar classification accuracy. These are multimodal multiobjective optimization (MMO) problems in feature selection. In this paper, the MMO problems in feature selection are described in detail. Then, the great significance and importance to find these different feature subsets are discussed. Two modified MMO algorithms are used to solve the MMO feature selection problems. Simulation results show that these MMO algorithms can find more feature subsets than unimodal optimization algorithms.

In sparse reconstruction, a small amount of samples are used to reconstruct sparse signal. This problem can be converted into multi-objective optimization problem, which considers sparsity and measurement error two competing cost function terms. Multi-objective optimization algorithm features unique advantage for solving this nonlinear and non-derivable problem. In order to trade off the sparsity and measurement error, in this paper, a novel multi-objective brainstorm algorithm based on objective space is used to optimize these two goals. Meanwhile, the local search ability of the algorithm is enhanced by introducing iterative half-thresholding operator in the frame of L1/2 regulation. On the basis of the Pareto solution set, the knee point is selected without prior signal information. In a coarse-to-fine manner, the algorithm has a certain tolerance to noise, and we give an analysis for this. According to the results on the eighteen test functions compared with several state-of-the-art compressive sensing recover methods, the effectiveness of multiobjective brainstorm algorithm based on objective space is verified for sparse optimization problems, which achieving higher recover and smaller error and reducing the average computational time for the long signal.

Some multi-objective evolutionary algorithms have been introduced to solve sparse optimization problems in recent years. These multi-objective sparse optimization algorithms obtain a set of solutions with different sparsities. However, for a specific sparse optimization problem, a unique sparse solution should be selected from the whole Pareto Set (PS). Usually, knee point in the PF is a preferred solution if the decision maker has no special preference. An effective knee point selection method plays a pivotal role in multi-objective sparse optimization. In this paper, a study on the knee point selection methods in multiobjective sparse optimization problems has been done. Three knee point selection methods, which are angle-based method, the weighted sum of objective values method and the distance to the extreme line method, are compared and the experimental results indicate that the second method is better than the others. Finally, an analysis of parameter in the best knee point selection method is conducted and an optimal setting range of parameters is given.

This paper presents a new particle swarm optimizer for solving multimodal multiobjective optimization problems which may have more than one Pareto-optimal solution corresponding to the same objective function value. The proposed method features an index-based ring topology to induce stable niches that allow the identification of a larger number of Pareto-optimal solutions, and adopts a special crowding distance concept as a density metric in the decision and objective spaces. The algorithm is shown to not only locate and maintain a larger number of Pareto-optimal solutions, but also to obtain good distributions in both the decision and objective spaces. In addition, new multimodal multiobjective optimization test functions and a novel performance indicator are designed for the purpose of assessing the performance of the proposed algorithms. An effectiveness validation study is carried out comparing the proposed method with five other algorithms using the benchmark functions to prove its effectiveness.

In real world applications, there are many multi-objective optimization problems. Most existing multi-objective optimization algorithms focus on improving the diversity, spread and convergence of the solutions in the objective space. Few works study the distribution of solutions in the decision space. In practical applications, some multi-objective problems have different Pareto sets with the same objective values and these problems are defined as multimodal multi-objective optimization problems. It is of great significance to provide all the Pareto sets for the decision maker. This paper describes the concept of multimodal multi-objective optimization problems in detail. Novel test functions are also designed to judge the performance of different algorithms. Moreover, some existing multi-objective algorithms are tested and compared. Finally, a decision space based niching multi-objective evolutionary algorithm is proposed to solve these problems. The experimental results suggest that existing multi-objective optimization algorithms fail to find all the Pareto sets while the proposed algorithm is able to find almost all the Pareto sets without deteriorating the distribution of solutions in the objective space.

Multimodal optimization amounts to finding multiple global and local optima (as opposed to a single solution) of a function, so that the user can have a better knowledge about different optimal solutions in the search space and when needed, the current solution may be switched to a more suitable one while still maintaining the optimal system performance. Niching particle swarm optimizers (PSOs) have been widely used by the evolutionary computation community for solving real-parameter multimodal optimization problems. However, most of the existing PSO-based niching algorithms are difficult to use in practice because of their poor local search ability and requirement of prior knowledge to specify certain niching parameters. This paper has addressed these issues by proposing a distance-based locally informed particle swarm (LIPS) optimizer, which eliminates the need to specify any niching parameter and enhance the fine search ability of PSO. Instead of using the global best particle, LIPS uses several local bests to guide the search of each particle. LIPS can operate as a stable niching algorithm by using the information provided by its neighborhoods. The neighborhoods are estimated in terms of Euclidean distance. The algorithm is compared with a number of state-of-the-art evolutionary multimodal optimizers on 30 commonly used multimodal benchmark functions. The experimental results suggest that the proposed technique is able to provide statistically superior and more consistent performance over the existing niching algorithms on the test functions, without incurring any severe computational burdens.

In this paper, Dynamic Multi-Swarm Particle Swarm Optimizer (DMS-PSO) which was first designed for solving single objective optimizations problems is extended to solve Multi-objective optimization problems with constraints. Through analysis, novel pbest and lbest updating criteria which are more suitable for solving Multi-objective optimization problems are proposed. By combining the external archive and the novel updating criteria, excellent performance is achieved by DMS-MO-PSO on eight benchmark test functions.

In this paper, a neighborhood mutation strategy is proposed and integrated with various niching differential evolution (DE) algorithms to solve multimodal optimization problems. Although variants of DE are highly effective in locating a single global optimum, no DE variant performs competitively when solving multi-optima problems. In the proposed neighborhood based differential evolution, the mutation is performed within each Euclidean neighborhood. The neighborhood mutation is able to maintain the multiple optima found during the evolution and evolve toward the respective global/local optimum. To test the performance of the proposed neighborhood mutation DE, a total of 29 problem instances are used. The proposed algorithms are compared with a number of state-of-the-art multimodal optimization approaches and the experimental results suggest that although the idea of neighborhood mutation is simple, it is able to provide better and more consistent performance over the state-of-the-art multimodal algorithms. In addition, a comparative survey on niching algorithms and their applications are also presented.

Constrained multi-objective differential evolution (CMODE) is a population-based stochastic search technique for solving constrained multi-objective optimization problems. Although CMODE is a powerful and efficient search algorithm, it frequently suffers from pre-mature convergence, especially when there are numerous local Pareto optimal solutions. In this paper, a diversity enhanced constrained multi-objective differential evolution (DE-CMODE) is proposed to overcome the pre-mature convergence problem. The performance of DE-MODE is evaluated on a set of 8 benchmark problems. As shown in the experimental results, the DE-CMODE performs either better or similar to the classical CMODE.

Multi-modal optimization refers to locating not only one optimum but a set of locally optimal solutions. Niching is an important technique to solve multi-modal optimization problems. The ability of discover and maintain multiple niches is the key capability of these algorithms. In this paper, differential evolution with an ensemble of restricted tournament selection (ERTS-DE) algorithm is introduced to perform multimodal optimization. The algorithms is tested on 15 newly designed scalable benchmark multi-modal optimization problems and compared with the crowding differential evolution (Crowding-DE) in the literature. As shown by the experimental results, the proposed algorithm outperforms the Crowding-DE on the novel scalable benchmark problems.

In this paper, the nonlinear constrained multi-objective environmental economic dispatch (EED) problem is solved using fast multi-objective evolutionary programming (FMOEP). Due to the global warming by fossil fuel, environmental concern becomes more and more important in recent years. The purpose of multi-objective optimization algorithm is minimizing all the different objectives simultaneously and finds the best tradeoff solution for this environmental/economic dispatch problem. In order to evaluate the performance of FMOEP on EED problems, the standard IEEE 30-bus six-generator test system is studied. The performance is compared against NSGAH and a number of results reported in literature. The results show that the FMOEP is effective in solving EED problems.

PID controller has been widely applied in engineering area. In this paper, multi-objective differential evolution (MODE) is used to design a multi-objective robust PID controller for two MIMO systems known as distillation column plant and longitudinal control system of the super maneuverable F18/HARV fighter aircraft. Multi-objective robust PID controller problem is formulated by minimizing integral squared error (ISE) and balanced robust performance criteria. The performance of the optimum PID controllers that obtain by MODE is compared with performance reported in literature by other methods in terms of the sum of ISE and balanced robust performance criteria. The results show that the PID controllers obtained by MODE can outperform various optimal PID controllers reported in literature.

Multi-objective differential evolution (MODE) is a powerful and efficient population-based stochastic search technique for solving multi-objective optimization problems in many scientific and engineering fields. However, premature convergence is the major drawback of MODE, especially when there are numerous local Pareto optimal solutions. To overcome this problem, we propose a MODE with a diversity enhancement (MODE-DE) mechanism to prevent the algorithm becoming trapped in a locally optimal Pareto front. The proposed algorithm combines the current population with a number of randomly generated parameter vectors to increase the diversity of the differential vectors and thereby the diversity of the newly generated offspring. The performance of the MODE-DE algorithm was evaluated on a set of 19 benchmark problem codes available from http://www3.ntu.edu.sg/home/epnsugan . With the proposed method, the performances were either better than or equal to those of the MODE without the diversity enhancement.

In Evolutionary Algorithms, the occurrence of premature convergence is due to lack of diversity in the population during the search process. The effect may be more predominant if the optimization problem includes constraints. In this paper we propose an explicit memory based diversity enhancement Adaptive Evolutionary Programming (DivEnh-AEP) method to solve constraint optimization problems of CEC 2006.

In this paper, multi-objective evolutionary programming (MOEP) using fuzzy rank-sum with diversified selection is introduced. The performances of this algorithm as well as MOEP with non-domination sorting on the set of benchmark functions provided for CEC2009 Special Session and competition on Multi-objective Optimization are reported. With this rank-sum sorting and diversified selection, the speed of the algorithm has increased significantly, in particular by about twenty times on five objective problems when compared with the implementation using the non-domination sorting. Beside this, the proposed approach has performed either comparable or better than the MOEP with non-domination sorting.