Search

When solving constrained optimization problems (COPs) by evolutionary algorithms, the search algorithm plays a crucial role. In general, we expect that the search algorithm has the capability to balance not only diversity and convergence but also constraints and objective function during the evolution. For this purpose, this paper proposes a composite differential evolution (DE) for constrained optimization, which includes three different trial vector generation strategies with distinct advantages. In order to strike a balance between diversity and convergence, one of these three trial vector generation strategies is able to increase diversity, and the other two exhibit the property of convergence. In addition, to accomplish the tradeoff between constraints and objective function, one of the two trial vector generation strategies for convergence is guided by the individual with the least degree of constraint violation in the population, and the other is guided by the individual with the best objective function value in the population. After producing offspring by the proposed composite DE, the feasibility rule and the $\boldsymbol {\varepsilon }$ constrained method are combined elaborately for selection in this paper. Moreover, a restart scheme is proposed to help the population jump out of a local optimum in the infeasible region for some extremely complicated COPs. By assembling the above techniques together, a constrained composite DE is proposed. The experiments on two sets of benchmark test functions with various features, i.e., 24 test functions from IEEE CEC2006 and 18 test functions with 10 dimensions and 30 dimensions from IEEE CEC2010, have demonstrated that the proposed method shows better or at least competitive performance against other state-of-the-art methods. [ABSTRACT FROM AUTHOR]

Decomposition-based many-objective evolutionary algorithms generally decompose the objective space into multiple subregions with the help of a set of reference vectors. The resulting subregions are fixed since the reference vectors are usually predefined. When the optimization problem has a complicated Pareto front (PF), this decomposition may decrease the algorithm performance. To deal with this problem, this paper proposes a dynamical decomposition strategy. Instead of using predefined reference vectors, solution themselves are used as reference vectors. Thus, they are adapted to the shape of PF automatically. Besides, the subregions are produced one by one through successively bipartitioning the objective space. The resulting subregions are not fixed but dynamically determined by the population solutions as well as the subregions produced previously. Based on this strategy, a solution ranking method, named dynamical-decomposition-based ranking method (DDR), is proposed which can be employed in the mating selection and environmental selection in commonly used algorithm frameworks. Compared with those in the other decomposition-based algorithms, DDR has the following properties: 1) no predefined reference vectors are required; 2) less parameters are involved; and 3) the ranking results can not only be utilized directly to select solutions but also serve as a secondary criterion in traditional Pareto-based algorithms. In this paper, DDR is equipped in two algorithm frameworks for handling many-objective optimization problems. Comparisons with five state-of-the-art algorithms on 31 widely used test problems are carried out to test the performance of the proposed approach. The experimental results have shown the effectiveness of the proposed approach in keeping a good tradeoff between convergence and diversity. [ABSTRACT FROM AUTHOR]

This paper develops a decomposition-based coevolutionary algorithm for many-objective optimization, which evolves a number of subpopulations in parallel for approaching the set of Pareto optimal solutions. The many-objective problem is decomposed into a number of subproblems using a set of well-distributed weight vectors. Accordingly, each subpopulation of the algorithm is associated with a weight vector and is responsible for solving the corresponding subproblem. The exploration ability of the algorithm is improved by using a mating pool that collects elite individuals from the cooperative subpopulations for breeding the offspring. In the subsequent environmental selection, the top-ranked individuals in each subpopulation, which are appraised by aggregation functions, survive for the next iteration. Two new aggregation functions with distinct characteristics are designed in this paper to enhance the population diversity and accelerate the convergence speed. The proposed algorithm is compared with several state-of-the-art many-objective evolutionary algorithms on a large number of benchmark instances, as well as on a real-world design problem. Experimental results show that the proposed algorithm is very competitive. [ABSTRACT FROM AUTHOR]

Most existing multiobjective evolutionary algorithms experience difficulties in solving many-objective optimization problems due to their incapability to balance convergence and diversity in the high-dimensional objective space. In this paper, we propose a novel many-objective evolutionary algorithm using a one-by-one selection strategy. The main idea is that in the environmental selection, offspring individuals are selected one by one based on a computationally efficient convergence indicator to increase the selection pressure toward the Pareto optimal front. In the one-by-one selection, once an individual is selected, its neighbors are de-emphasized using a niche technique to guarantee the diversity of the population, in which the similarity between individuals is evaluated by means of a distribution indicator. In addition, different methods for calculating the convergence indicator are examined and an angle-based similarity measure is adopted for effective evaluations of the distribution of solutions in the high-dimensional objective space. Moreover, corner solutions are utilized to enhance the spread of the solutions and to deal with scaled optimization problems. The proposed algorithm is empirically compared with eight state-of-the-art many-objective evolutionary algorithms on 80 instances of 16 benchmark problems. The comparative results demonstrate that the overall performance of the proposed algorithm is superior to the compared algorithms on the optimization problems studied in this paper. [ABSTRACT FROM PUBLISHER]

While Pareto-based multiobjective optimization algorithms continue to show effectiveness for a wide range of practical problems that involve mostly two or three objectives, their limited application for many-objective problems, due to the increasing proportion of nondominated solutions and the lack of sufficient selection pressure, has also been gradually recognized. In this paper, we revive an early developed and computationally expensive strength Pareto-based evolutionary algorithm by introducing an efficient reference direction-based density estimator, a new fitness assignment scheme, and a new environmental selection strategy, for handling both multiobjective and many-objective problems. The performance of the proposed algorithm is validated and compared with some state-of-the-art algorithms on a number of test problems. Experimental studies demonstrate that the proposed method shows very competitive performance on both multiobjective and many-objective problems considered in this paper. Besides, our extensive investigations and discussions reveal an interesting finding, that is, diversity-first-and-convergence-second selection strategies may have great potential to deal with many-objective optimization. [ABSTRACT FROM AUTHOR]

This paper compares three modern and two classical multiobjective optimizers (MOOs) as applied to real-world problems in electromagnetics. The behavior of sophisticated optimizers on simple test functions has been studied exhaustively. In contrast, the algorithms here are tested on practical applications, where the function evaluations are computationally expensive, making the convergence rate a crucial factor. The examples considered include the optimization of a narrowband slot antenna, a mushroom-type electromagnetic bandgap structure, and an ultrawideband Vivaldi antenna. Another popular topic in the literature is in comparing classical MOOs on electromagnetics problems. The modern optimizers chosen in this paper are state of the art and each has a distinct design philosophy. This paper introduces two unique MOOs to the electromagnetics community: BORG, an auto-adaptive genetic algorithm and the Multi-Objective Covariance Matrix Adaptation Evolutionary Strategy (MO-CMA-ES), an extension of the popular single-objective CMA-ES. These algorithms are compared to the Multi-objective Evolutionary Algorithm based on Decomposition (MOEA/D), a Chebysheff scalarization algorithm, and two classical MOOs. This paper will study the behavior of these algorithms on problems in electromagnetics with a limited number of function evaluations using five distinct metrics and will provide useful guidelines and recommended optimizer settings. [ABSTRACT FROM AUTHOR]

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

With the increase in the number of optimization objectives, balancing the convergence and diversity in evolutionary multiobjective optimization becomes more intractable. So far, a variety of evolutionary algorithms have been proposed to solve many-objective optimization problems (MaOPs) with more than three objectives. Most of the existing algorithms, however, find difficulties in simultaneously counterpoising convergence and diversity during the whole evolutionary process. To address the issue, this paper proposes to solve MaOPs via multistage evolutionary search. To be specific, a two-stage evolutionary algorithm is developed, where the convergence and diversity are highlighted during different search stages to avoid the interferences between them. The first stage pushes multiple subpopulations with different weight vectors to converge to different areas of the Pareto front. After that, the nondominated solutions coming from each subpopulation are selected for generating a new population for the second stage. Moreover, a new environmental selection strategy is designed for the second stage to balance the convergence and diversity close to the Pareto front. This selection strategy evenly divides each objective dimension into a number of intervals, and then one solution having the best convergence in each interval will be retained. To assess the performance of the proposed algorithm, 48 benchmark functions with 7, 10, and 15 objectives are used to make comparisons with five representative many-objective optimization algorithms. [ABSTRACT FROM AUTHOR]

Balancing population diversity and convergence is critical for evolutionary algorithms to solve many-objective optimization problems (MaOPs). In this paper, a two-round environmental selection strategy is proposed to pursue good tradeoff between population diversity and convergence for many-objective evolutionary algorithms (MaOEAs). Particularly, in the first round, the solutions with small neighborhood density are picked out to form a candidate pool, where the neighborhood density of a solution is calculated based on a novel adaptive position transformation strategy. In the second round, the best solution in terms of convergence is selected from the candidate pool and inserted into the next generation. The procedure is repeated until a new population is generated. The two-round selection strategy is embedded into an MaOEA framework and the resulting algorithm, namely, 2REA, is compared with eight state-of-the-art MaOEAs on various benchmark MaOPs. The experimental results show that 2REA is very competitive with the compared MaOEAs and the two-round selection strategy works well on balancing population diversity and convergence. [ABSTRACT FROM AUTHOR]

In this paper, a novel particle swarm optimization (PSO) algorithm is put forward where a sigmoid-function-based weighting strategy is developed to adaptively adjust the acceleration coefficients. The newly proposed adaptive weighting strategy takes into account both the distances from the particle to the global best position and from the particle to its personal best position, thereby having the distinguishing feature of enhancing the convergence rate. Inspired by the activation function of neural networks, the new strategy is employed to update the acceleration coefficients by using the sigmoid function. The search capability of the developed adaptive weighting PSO (AWPSO) algorithm is comprehensively evaluated via eight well-known benchmark functions including both the unimodal and multimodal cases. The experimental results demonstrate that the designed AWPSO algorithm substantially improves the convergence rate of the particle swarm optimizer and also outperforms some currently popular PSO algorithms. [ABSTRACT FROM AUTHOR]

Many-objective optimization has posed a great challenge to the classical Pareto dominance-based multiobjective evolutionary algorithms (MOEAs). In this paper, an evolutionary algorithm based on a new dominance relation is proposed for many-objective optimization. The proposed evolutionary algorithm aims to enhance the convergence of the recently suggested nondominated sorting genetic algorithm III by exploiting the fitness evaluation scheme in the MOEA based on decomposition, but still inherit the strength of the former in diversity maintenance. In the proposed algorithm, the nondominated sorting scheme based on the introduced new dominance relation is employed to rank solutions in the environmental selection phase, ensuring both convergence and diversity. The proposed algorithm is evaluated on a number of well-known benchmark problems having 3–15 objectives and compared against eight state-of-the-art algorithms. The extensive experimental results show that the proposed algorithm can work well on almost all the test functions considered in this paper, and it is compared favorably with the other many-objective optimizers. Additionally, a parametric study is provided to investigate the influence of a key parameter in the proposed algorithm. [ABSTRACT FROM PUBLISHER]

Differential evolution is one of the most powerful evolutionary algorithms for single objective optimization problems in the literature. Its application in the multiobjective optimization problems is also very successful, and many kinds of promising multiobjective differential evolution (MODE) algorithms have been proposed in the literature. This paper develops a new variant of MODE with two features. First, a set of personal archives are maintained to evolve the search process instead of a population with a fixed size, and a truncation procedure is used to enhance selection pressure. Second, multiple biased mutation operators incorporating the target solution quality are proposed, and an adaptive selection method is adopted to allocate the mutation operators to solutions. The proposed MODE is referred to as the MODE with personal archive and biased self-adaptive mutation selection (BiasMOSaDE). A set of 31 benchmark multiobjective problems selected from the literature are adopted to evaluate its performance. Computational results illustrate that the proposed BiasMOSaDE is competitive or even superior to several state-of-the-art MODEs in the literature. [ABSTRACT FROM AUTHOR]

A variety of evolutionary algorithms have been proposed for many-objective optimization in recent years. However, the difficulties in balancing the convergence and diversity of the population and selecting promising parents for offspring reproduction remain. In this paper, we propose a subregion division-based evolutionary algorithm with an effective mating selection strategy, termed SdEA, for many-objective optimization. In SdEA, a subregion division approach is proposed to divide the objective space into different subregions for balancing the diversity and convergence of the population. Besides, an effective mating selection strategy is proposed to enhance the diversity of the mating pool solutions, aimed at enhancing the selection probability of solutions in the sparse subregions. The proposed SdEA is compared with five state-of-the-art many-objective evolutionary algorithms on 23 test problems from DTLZ, WFG, and MaF test suites. Experimental results on these problems demonstrate that the proposed algorithm is competitive in solving many-objective problems. Furthermore, the proposed mating selection strategy is embedded in several evolutionary algorithms and experimental results demonstrate its effectiveness on improving the performance of the embedded algorithms. [ABSTRACT FROM AUTHOR]

Decomposition-based evolutionary algorithms have been quite successful in solving optimization problems involving two and three objectives. Recently, there have been some attempts to exploit the strengths of decomposition-based approaches to deal with many objective optimization problems. Performance of such approaches are largely dependent on three key factors: 1) means of reference point generation; 2) schemes to simultaneously deal with convergence and diversity; and 3) methods to associate solutions to reference directions. In this paper, we introduce a decomposition-based evolutionary algorithm wherein uniformly distributed reference points are generated via systematic sampling, balance between convergence and diversity is maintained using two independent distance measures, and a simple preemptive distance comparison scheme is used for association. In order to deal with constraints, an adaptive epsilon formulation is used. The performance of the algorithm is evaluated using standard benchmark problems, i.e., DTLZ1-DTLZ4 for 3, 5, 8, 10, and 15 objectives, WFG1-WFG9, the car side impact problem, the water resource management problem, and the constrained ten-objective general aviation aircraft design problem. Results of problems involving redundant objectives and disconnected Pareto fronts are also included in this paper to illustrate the capability of the algorithm. The study clearly highlights that the proposed algorithm is better or at par with recent reference direction-based approaches for many objective optimization. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Neighborhood information plays an important role in improving the performance of evolutionary computation in various optimization scenarios, particularly in the context of multimodal optimization. Several neighborhood concepts, i.e., index-based neighborhood, nearest neighborhood, and fuzzy neighborhood, have been studied and engaged in the design of niching methods. However, the use of these neighborhood concepts requires the specification of some problem-related parameters, which is difficult to determine without a prior knowledge. In this paper, we introduce a new neighborhood concept based on a geometrical construction called Voronoi diagram. The new concept offers two advantages at the expense of increasing the computational complexity to a higher level. It eliminates the need of additional parameters and it is more informative than the existing ones. The information provided by the Voronoi neighbors of an individual can be exploited to estimate the evolutionary state. Based on the information, we divide the population into three groups and assign each group a different reproduction strategy to support the exploration and exploitation of the search space. We show the use of the concept in the design of an effective evolutionary algorithm for multimodal optimization. The experiments have been conducted to investigate the performance of the algorithm. The results reveal that the proposed algorithm compare favorably with the state-of-the-art algorithms designed based on other types of neighborhood concepts. [ABSTRACT FROM AUTHOR]

The decomposition-based evolutionary algorithm has become an increasingly popular choice for posterior multiobjective optimization. Facing the challenges of an increasing number of objectives, many techniques have been developed which help to balance the convergence and diversity. Nevertheless, according to a recent study by Ishibuchi et al., due to the predefined search directions toward the ideal point, their performance strongly depends on the Pareto front (PF) shapes, especially the orientation of the PFs. To balance the convergence and diversity for decomposition-based methods and to alleviate their performance dependence on the orientation of the PFs, this paper develops an adversarial decomposition method for many-objective optimization, which leverages the complementary characteristics of different subproblem formulations within a single paradigm. More specifically, two populations are co-evolved by two subproblem formulations with different contours and adversarial search directions. To avoid allocating redundant computational resources to the same region of the PF, the two populations are matched into one-to-one solution pairs according to their working regions upon the PF. Each solution pair can at most contribute one principal mating parent during the mating selection process. When comparing nine state-of-the-art many-objective optimizers, we have witnessed the competitive performance of our proposed algorithm on 130 many-objective test problems with various characteristics, including regular and inverted PFs. [ABSTRACT FROM AUTHOR]

Decomposition-based evolutionary algorithms have shown great potential in many-objective optimization. However, the lack of theoretical studies on decomposition methods has hindered their further development and application. In this paper, we first theoretically prove that weight sum, Tchebycheff, and penalty boundary intersection decomposition methods are essentially interconnected. Inspired by this, we further show that highly customized dominance relationship can be derived from decomposition for any given decomposition vector. A new evolutionary algorithm is then proposed by applying the customized dominance relationship with adaptive strategy to each subpopulation of multiobjective to multiobjective framework. Experiments are conducted to compare the proposed algorithm with five state-of-the-art decomposition-based evolutionary algorithms on a set of well-known scaled many-objective test problems with 5 to 15 objectives. Simulation results have shown that the proposed algorithm can make better use of the decomposition vectors to achieve better performance. Further investigations on unscaled many-objective test problems verify the robust and generality of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Convergence and diversity are interdependently handled during the evolutionary process by most existing many-objective evolutionary algorithms (MaOEAs). In such a design, the degraded performance of one would deteriorate the other, and only solutions with both are able to improve the performance of MaOEAs. Unfortunately, it is not easy to constantly maintain a population of solutions with both convergence and diversity. In this paper, an MaOEA based on two independent stages is proposed for effectively solving many-objective optimization problems (MaOPs), where the convergence and diversity are addressed in two independent and sequential stages. To achieve this, we first propose a nondominated dynamic weight aggregation method by using a genetic algorithm, which is capable of finding the Pareto-optimal solutions for MaOPs with concave, convex, linear and even mixed Pareto front shapes, and then these solutions are employed to learn the Pareto-optimal subspace for the convergence. Afterward, the diversity is addressed by solving a set of single-objective optimization problems with reference lines within the learned Pareto-optimal subspace. To evaluate the performance of the proposed algorithm, a series of experiments are conducted against six state-of-the-art MaOEAs on benchmark test problems. The results show the significantly improved performance of the proposed algorithm over the peer competitors. In addition, the proposed algorithm can focus directly on a chosen part of the objective space if the preference area is known beforehand. Furthermore, the proposed algorithm can also be used to effectively find the nadir points. [ABSTRACT FROM AUTHOR]

There have been few researches on solving multimodal multiobjective optimization problems, whereas they are commonly seen in real-world applications but difficult for the existing evolutionary optimizers. In this paper, we propose a novel multimodal multiobjective evolutionary algorithm using two-archive and recombination strategies. In the proposed algorithm, the properties of decision variables and the relationships among them are analyzed at first to guide the evolutionary search. Then, a general framework using two archives, i.e., the convergence and the diversity archives, is adopted to cooperatively solve these problems. Moreover, the diversity archive simultaneously employs a clustering strategy to guarantee diversity in the objective space and a niche-based clearing strategy to promote the same in the decision space. At the end of evolution process, solutions in the convergence and the diversity archives are recombined to obtain a large number of multiple Pareto optimal solutions. In addition, a set of benchmark test functions and a performance metric are designed for multimodal multiobjective optimization. The proposed algorithm is empirically compared with two state-of-the-art evolutionary algorithms on these test functions. The comparative results demonstrate that the overall performance of the proposed algorithm is significantly superior to the competing algorithms. [ABSTRACT FROM AUTHOR]

Existing multiobjective evolutionary algorithms (MOEAs) perform well on multiobjective optimization problems (MOPs) with regular Pareto fronts in which the Pareto optimal solutions distribute continuously over the objective space. When the Pareto front is discontinuous or degenerated, most existing algorithms cannot achieve good results. To remedy this issue, a clustering-based adaptive MOEA (CA-MOEA) is proposed in this paper for solving MOPs with irregular Pareto fronts. The main idea is to adaptively generate a set of cluster centers for guiding selection at each generation to maintain diversity and accelerate convergence. We investigate the performance of CA-MOEA on 18 widely used benchmark problems. Our results demonstrate the competitiveness of CA-MOEA for multiobjective optimization, especially for problems with irregular Pareto fronts. In addition, CA-MOEA is shown to perform well on the optimization of the stretching parameters in the carbon fiber formation process. [ABSTRACT FROM AUTHOR]

In this paper, a novel multiobjective evolutionary algorithm (MOEA/CT) is proposed to better manage convergence and distribution of solutions when MOEAs are used for solving multiobjective optimization problems. The coordinate transformation strategy, an external archive update strategy, and a diversity maintenance strategy are proposed in MOEA/CT. The coordinate transformation strategy in the objective space is designed to find more efficient solutions that can accelerate the convergence process. Based on the coordinate transformation strategy, a novel update strategy and diversity maintenance approach for selecting nondominated solutions from the external archive set are integrated in MOEA/CT for getting better distribution of the solutions. The proposed MOEA/CT is compared with eight state-of-art algorithms on six biobjective and seven tri-objective test problems. In terms of four performance metrics, the comparative experimental results demonstrate that MOEA/CT outperforms the other eight competitors and it can achieve solutions with better distribution and better convergence to the Pareto front. In addition, parameter sensitivity analysis is provided to investigate the effect of a key parameter in MOEA/CT; the proposed three strategies are also studied individually to investigate their contribution to MOEA/CT; the performance analysis along with the capacity of external archive is given to clearly make the influence in MOEA/CT; finally, the scalability performance of MOEA/CT is investigated and compared with five notable many-objective evolutionary algorithms on the DTLZ and WFG test suites with 5, 8, 10, and 15 objectives. [ABSTRACT FROM AUTHOR]

The performance of traditional multiobjective evolutionary algorithms (MOEAs) often deteriorates rapidly as the number of decision variables increases. While some efforts were made to design new algorithms by adapting existing techniques to large-scale single-objective optimization to the MOEA context, the specific difficulties that may arise from large-scale multiobjective optimization have rarely been studied. In this paper, the exclusive challenges along with the increase of the number of variables of a multiobjective optimization problem (MOP) are examined empirically, and the popular benchmarks are categorized into three groups accordingly. Problems in the first category only require MOEAs to have stronger convergence, and can thus be mitigated using techniques employed in large-scale single-objective optimization. Problems that require MOEAs to have stronger diversification but ignore a correlation between position and distance functions are grouped as the second. The rest of the problems that pose a great challenge to the balance between diversification and convergence by considering a correlation between position and distance functions are grouped as the third. While existing large-scale MOEAs perform well on the problems in the first two categories, they suffer a significant loss when applied to those in the third category. To solve large-scale MOPs in this category, we have developed a novel indicator-based algorithm with an enhanced diversification mechanism. The proposed algorithm incorporates a new solution generator with an external archive, thus forcing the search toward different subregions of the Pareto front using a dual local search mechanism. The results obtained by applying the proposed algorithm to a wide variety of problems (108 instances in total) with up to 8192 variables demonstrate that it outperforms eight state-of-the-art approaches on the examined problems in the third category and show its advantage in the balance between diversification and convergence. [ABSTRACT FROM AUTHOR]

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

In decomposition-based multiobjective evolutionary algorithms, the setting of search directions (or weight vectors), and the choice of reference points (i.e., the ideal point or the nadir point) in scalarizing functions, are of great importance to the performance of the algorithms. This paper proposes a new decomposition-based many-objective optimizer by simultaneously using adaptive search directions and two reference points. For each parent, binary search directions are constructed by using its objective vector and the two reference points. Each individual is simultaneously evaluated on two fitness functions—which are motivated by scalar projections—that are deduced to be the differences between two penalty-based boundary intersection (PBI) functions, and two inverted PBI functions, respectively. Solutions with the best value on each fitness function are emphasized. Moreover, an angle-based elimination procedure is adopted to select diversified solutions for the next generation. The use of adaptive search directions aims at effectively handling problems with irregular Pareto-optimal fronts, and the philosophy of using the ideal and nadir points simultaneously is to take advantages of the complementary effects of the two points when handling problems with either concave or convex fronts. The performance of the proposed algorithm is compared with seven state-of-the-art multi-/many-objective evolutionary algorithms on 32 test problems with up to 15 objectives. It is shown by the experimental results that the proposed algorithm is flexible when handling problems with different types of Pareto-optimal fronts, obtaining promising results regarding both the quality of the returned solution set and the efficiency of the new algorithm. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

When solving constrained multiobjective optimization problems, an important issue is how to balance convergence, diversity, and feasibility simultaneously. To address this issue, this paper proposes a parameter-free constraint handling technique, a two-archive evolutionary algorithm, for constrained multiobjective optimization. It maintains two collaborative archives simultaneously: one, denoted as the convergence-oriented archive (CA), is the driving force to push the population toward the Pareto front; the other one, denoted as the diversity-oriented archive (DA), mainly tends to maintain the population diversity. In particular, to complement the behavior of the CA and provide as much diversified information as possible, the DA aims at exploring areas under-exploited by the CA including the infeasible regions. To leverage the complementary effects of both archives, we develop a restricted mating selection mechanism that adaptively chooses appropriate mating parents from them according to their evolution status. Comprehensive experiments on a series of benchmark problems and a real-world case study fully demonstrate the competitiveness of our proposed algorithm, in comparison to five state-of-the-art constrained evolutionary multiobjective optimizers. [ABSTRACT FROM AUTHOR]

Evolutionary algorithms have shown their promise in coping with many-objective optimization problems. However, the strategies of balancing convergence and diversity and the effectiveness of handling problems with irregular Pareto fronts (PFs) are still far from perfect. To address these issues, this paper proposes an adaptive sorting-based evolutionary algorithm based on the idea of decomposition. First, we propose an adaptive sorting-based environmental selection strategy. Solutions in each subpopulation (partitioned by reference vectors) are sorted based on their convergence. Those with better convergence are further sorted based on their diversity, then being selected according to their sorting levels. Second, we provide an adaptive promising subpopulation sorting-based environmental selection strategy for problems which may have irregular PFs. This strategy provides additional sorting-based selection effort on promising subpopulations after the general environmental selection process. Third, we extend the algorithm to handle constraints. Finally, we conduct an extensive experimental study on the proposed algorithm by comparing with start-of-the-state algorithms. Results demonstrate the superiority of the proposed algorithm. [ABSTRACT FROM AUTHOR]

Conventional evolutionary algorithms (EAs) are not well suited for solving expensive optimization problems due to the fact that they often require a large number of fitness evaluations to obtain acceptable solutions. To alleviate the difficulty, this paper presents a multitasking evolutionary optimization framework for solving computationally expensive problems. In the framework, knowledge is transferred from a number of computationally cheap optimization problems to help the solution of the expensive problem on the basis of the recently proposed multifactorial EA (MFEA), leading to a faster convergence of the expensive problem. However, existing MFEAs do not work well in solving multitasking problems whose optimums do not lie in the same location or when the dimensions of the decision space are not the same. To address the above issues, the existing MFEA is generalized by proposing two strategies, one for decision variable translation and the other for decision variable shuffling, to facilitate knowledge transfer between optimization problems having different locations of the optimums and different numbers of decision variables. To assess the effectiveness of the generalized MFEA (G-MFEA), empirical studies have been conducted on eight multitasking instances and eight test problems for expensive optimization. The experimental results demonstrate that the proposed G-MFEA works more efficiently for multitasking optimization and successfully accelerates the convergence of expensive optimization problems compared to single-task optimization. [ABSTRACT FROM AUTHOR]

In this paper, a decomposition-based artificial bee colony (ABC) algorithm is proposed to handle many-objective optimization problems (MaOPs). In the proposed algorithm, an MaOP is converted into a number of subproblems which are simultaneously optimized by a modified ABC algorithm. The hybrid of the decomposition-based algorithm and the ABC algorithm can make full use of the advantages of both algorithms. The former, with the help of a set of weight vectors, is able to maintain a good diversity among solutions, while the latter, with a fast convergence speed, is highly effective when solving a scalar optimization problem. Therefore, the convergence and diversity would be well balanced in the new algorithm. Moreover, subproblems in the proposed algorithm are handled unequally, and computational resources are dynamically allocated through specially designed onlooker bees and scout bees. The proposed algorithm is compared with five state-of-the-art many-objective evolutionary algorithms on 13 test problems with up to 50 objectives. It is shown by the experimental results that the proposed algorithm performs better than or comparably to other algorithms in terms of both quality of the final solution set and efficiency of the algorithms. Finally, as shown by the Wilcoxon signed-rank test results, the onlooker bees and scout bees indeed contribute to performance improvements of the algorithm. Given the high quality of solutions and the rapid running speed, the proposed algorithm could be a promising tool when approximating a set of well-converged and properly distributed nondominated solutions for MaOPs. [ABSTRACT FROM AUTHOR]

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

Multiobjective evolutionary algorithms (MOEAs) have proven their effectiveness and efficiency in solving problems with two or three objectives. However, recent studies show that MOEAs face many difficulties when tackling problems involving a larger number of objectives as their behavior becomes similar to a random walk in the search space since most individuals are nondominated with respect to each other. Motivated by the interesting results of decomposition-based approaches and preference-based ones, we propose in this paper a new decomposition-based dominance relation to deal with many-objective optimization problems and a new diversity factor based on the penalty-based boundary intersection method. Our reference point-based dominance (RP-dominance), has the ability to create a strict partial order on the set of nondominated solutions using a set of well-distributed reference points. The RP-dominance is subsequently used to substitute the Pareto dominance in nondominated sorting genetic algorithm-II (NSGA-II). The augmented MOEA, labeled as RP-dominance-based NSGA-II, has been statistically demonstrated to provide competitive and oftentimes better results when compared against four recently proposed decomposition-based MOEAs on commonly-used benchmark problems involving up to 20 objectives. In addition, the efficacy of the algorithm on a realistic water management problem is showcased. [ABSTRACT FROM AUTHOR]

Decomposition-based multiobjective evolutionary algorithms (MOEAs) have received increasing research interests due to their high performance for solving multiobjective optimization problems. However, scalarizing functions (SFs), which play a crucial role in balancing diversity and convergence in these kinds of algorithms, have not been fully investigated. This paper is mainly devoted to presenting two new SFs and analyzing their effect in decomposition-based MOEAs. Additionally, we come up with an efficient framework for decomposition-based MOEAs based on the proposed SFs and some new strategies. Extensive experimental studies have demonstrated the effectiveness of the proposed SFs and algorithm. [ABSTRACT FROM PUBLISHER]

The balance between convergence and diversity is the cornerstone of evolutionary multiobjective optimization (EMO). The recently proposed stable matching-based selection provides a new perspective to handle this balance under the framework of decomposition multiobjective optimization. In particular, the one-one stable matching between subproblems and solutions, which achieves an equilibrium between their mutual preferences, is claimed to strike a balance between convergence and diversity. However, the original stable marriage model has a high risk of matching a solution with an unfavorable subproblem, which finally leads to an imbalanced selection result. In this paper, we introduce the concept of incomplete preference lists into the stable matching model to remedy the loss of population diversity. In particular, each solution is only allowed to maintain a partial preference list consisting of its favorite subproblems. We implement two versions of stable matching-based selection mechanisms with incomplete preference lists: one achieves a two-level one-one matching and the other obtains a many-one matching. Furthermore, an adaptive mechanism is developed to automatically set the length of the incomplete preference list for each solution according to its local competitiveness. The effectiveness and competitiveness of our proposed methods are validated and compared with several state-of-the-art EMO algorithms on 62 benchmark problems. [ABSTRACT FROM PUBLISHER]

Metaheuristics have yielded very promising results for the frequency assignment problem (FAP). However, the results obtainable using currently published methods are far from ideal in complex, large-scale instances. This paper applies and extends some of the most recent advances in evolutionary algorithms to two common variants of the FAP, and shows how, in traditional techniques, two common issues affect their performance: 1) premature convergence and 2) the way in which neutral networks are handled. A recent replacement-based diversity management strategy is successfully applied to alleviate the premature convergence drawback. Additionally, by properly defining a distance metric, the performance in the presence of neutrality can also be greatly improved. The replacement strategy combines the principle of transforming a single-objective problem into a multiobjective one by considering diversity as an additional objective, with the idea of adapting the balance induced between exploration and exploitation to the requirements of the different optimization stages. Tests with 44 publicly available instances yield very competitive results. New best-known frequency plans were generated for 11 instances, whereas in the remaining ones the best-known solutions were replicated. Comparisons with a large number of strategies designed to delay convergence of the population clearly show the advantages of our novel proposals. [ABSTRACT FROM PUBLISHER]

Decomposition is a well-known strategy in traditional multiobjective optimization. However, the decomposition strategy was not widely employed in evolutionary multiobjective optimization until Zhang and Li proposed multiobjective evolutionary algorithm based on decomposition (MOEA/D) in 2007. MOEA/D proposed by Zhang and Li decomposes a multiobjective optimization problem into a number of scalar optimization subproblems and optimizes them in a collaborative manner using an evolutionary algorithm (EA). Each subproblem is optimized by utilizing the information mainly from its several neighboring subproblems. Since the proposition of MOEA/D in 2007, decomposition-based MOEAs have attracted significant attention from the researchers. Investigations have been undertaken in several directions, including development of novel weight vector generation methods, use of new decomposition approaches, efficient allocation of computational resources, modifications in the reproduction operation, mating selection and replacement mechanism, hybridizing decomposition- and dominance-based approaches, etc. Furthermore, several attempts have been made at extending the decomposition-based framework to constrained multiobjective optimization, many-objective optimization, and incorporate the preference of decision makers. Additionally, there have been many attempts at application of decomposition-based MOEAs to solve complex real-world optimization problems. This paper presents a comprehensive survey of the decomposition-based MOEAs proposed in the last decade. [ABSTRACT FROM AUTHOR]

Selection strategy, including mating selection and environmental selection, is a key ingredient in the design of evolutionary multiobjective optimization algorithms. Existing approaches, which have shown competitive performance in low-dimensional multiobjective optimization problems with two or three objectives, often encounter considerable challenges in many-objective optimization, where the number of objectives exceeds 3. This paper first provides a comprehensive analysis on the selection strategies in the current evolutionary many-objective optimization algorithms. Afterward, we propose a coordinated selection strategy to improve the performance of evolutionary algorithms in many-objective optimization. This selection strategy considers three crucial factors: 1) the new mating selection criterion considers both the quality of each selected parent and the effectiveness of the combination of selected parents; 2) the new environmental selection criterion directly focuses on the performance of the whole population rather than single individual alone; and 3) both selection steps are complement to each other and the coordination between them in the evolutionary process can achieve a better performance than each of them used individually. Furthermore, in order to handle the curse of dimensionality in many-objective optimization problems, a new convergence measure by distance and a new diversity measure by angle are developed in both selection steps. Experimental results on both DTLZ and WFG benchmark functions demonstrate the superiority of the proposed algorithm in comparison with six state-of-the-art designs in terms of both solution quality and computational efficiency. [ABSTRACT FROM AUTHOR]

Taking both convergence and diversity into consideration, this paper suggests a vector angle-based evolutionary algorithm for unconstrained (with box constraints only) many-objective optimization problems. In the proposed algorithm, the maximum-vector-angle-first principle is used in the environmental selection to guarantee the wideness and uniformity of the solution set. With the help of the worse-elimination principle, worse solutions in terms of the convergence (measured by the sum of normalized objectives) are allowed to be conditionally replaced by other individuals. Therefore, the selection pressure toward the Pareto-optimal front is strengthened. The proposed method is compared with other four state-of-the-art many-objective evolutionary algorithms on a number of unconstrained test problems with up to 15 objectives. The experimental results have shown the competitiveness and effectiveness of our proposed algorithm in keeping a good balance between convergence and diversity. Furthermore, it was shown by the results on two problems from practice (with irregular Pareto fronts) that our method significantly outperforms its competitors in terms of both the convergence and diversity of the obtained solution sets. Notably, the new algorithm has the following good properties: 1) it is free from a set of supplied reference points or weight vectors; 2) it has less algorithmic parameters; and 3) the time complexity of the algorithm is low. Given both good performance and nice properties, the suggested algorithm could be an alternative tool when handling optimization problems with more than three objectives. [ABSTRACT FROM PUBLISHER]

Traditional multiobjective evolutionary algorithms face a great challenge when dealing with many objectives. This is due to a high proportion of nondominated solutions in the population and low selection pressure toward the Pareto front. In order to tackle this issue, a series of indicator-based algorithms have been proposed to guide the search process toward the Pareto front. However, a single indicator might be biased and lead the population to converge to a subregion of the Pareto front. In this paper, a multi-indicator-based algorithm is proposed for many-objective optimization problems. The proposed algorithm, namely stochastic ranking-based multi-indicator Algorithm (SRA), adopts the stochastic ranking technique to balance the search biases of different indicators. Empirical studies on a large number (39 in total) of problem instances from two well-defined benchmark sets with 5, 10, and 15 objectives demonstrate that SRA performs well in terms of inverted generational distance and hypervolume metrics when compared with state-of-the-art algorithms. Empirical studies also reveal that, in the case a problem requires the algorithm to have strong convergence ability, the performance of SRA can be further improved by incorporating a direction-based archive to store well-converged solutions and maintain diversity. [ABSTRACT FROM PUBLISHER]

In evolutionary multiobjective optimization, maintaining a good balance between convergence and diversity is particularly crucial to the performance of the evolutionary algorithms (EAs). In addition, it becomes increasingly important to incorporate user preferences because it will be less likely to achieve a representative subset of the Pareto-optimal solutions using a limited population size as the number of objectives increases. This paper proposes a reference vector-guided EA for many-objective optimization. The reference vectors can be used not only to decompose the original multiobjective optimization problem into a number of single-objective subproblems, but also to elucidate user preferences to target a preferred subset of the whole Pareto front (PF). In the proposed algorithm, a scalarization approach, termed angle-penalized distance, is adopted to balance convergence and diversity of the solutions in the high-dimensional objective space. An adaptation strategy is proposed to dynamically adjust the distribution of the reference vectors according to the scales of the objective functions. Our experimental results on a variety of benchmark test problems show that the proposed algorithm is highly competitive in comparison with five state-of-the-art EAs for many-objective optimization. In addition, we show that reference vectors are effective and cost-efficient for preference articulation, which is particularly desirable for many-objective optimization. Furthermore, a reference vector regeneration strategy is proposed for handling irregular PFs. Finally, the proposed algorithm is extended for solving constrained many-objective optimization problems. [ABSTRACT FROM PUBLISHER]

Convergence performance and parametric sensitivity are two issues that tend to be neglected when extending differential evolution (DE) to multiobjective optimization (MO). To fill this research gap, we develop two novel mutation operators and a new parameter adaptation mechanism. A multiobjective DE variant is obtained through integration of the proposed strategies. The main innovation of this paper is the simultaneous use of individuals across generations from an objective-based perspective. Good convergence–diversity tradeoff and satisfactory exploration–exploitation balance are achieved via the hybrid cross-generation mutation operation. Furthermore, the cross-generation adaptation mechanism enables the individuals to self-adapt their associated parameters not only optimization-stage-wise but also objective-space-wise. Empirical results indicate the statistical superiority of the proposed algorithm over several state-of-the-art evolutionary algorithms in handling MO problems. [ABSTRACT FROM PUBLISHER]

Evolutionary algorithms have been successfully applied for exploring both converged and diversified approximate Pareto-optimal fronts in multiobjective optimization problems, two- or three-objective in general. However, when solving problems with many objectives, nearly all algorithms perform poorly due to the loss of selection pressure in fitness evaluation. An extremely large objective space could inadvertently deteriorate the effect of an evolutionary operator. In this paper, we propose a new approach to directly handle the challenges to solve many-objective optimization problems (MaOPs). This novel design includes two stages: first, the whole population quickly approaches a small number of “target” points near the true Pareto front; then, the proposed diversity improvement strategy is applied to facilitate these individuals to spread and well distribute. As a case study, the proposed algorithm based on this design is compared with five state-of-the-art algorithms. Experimental results show that the proposed method exhibits improved performance in both convergence and diversity for solving MaOPs. [ABSTRACT FROM PUBLISHER]

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. [ABSTRACT FROM AUTHOR]

Achieving balance between convergence and diversity is a key issue in evolutionary multiobjective optimization. Most existing methodologies, which have demonstrated their niche on various practical problems involving two and three objectives, face significant challenges in many-objective optimization. This paper suggests a unified paradigm, which combines dominance- and decomposition-based approaches, for many-objective optimization. Our major purpose is to exploit the merits of both dominance- and decomposition-based approaches to balance the convergence and diversity of the evolutionary process. The performance of our proposed method is validated and compared with four state-of-the-art algorithms on a number of unconstrained benchmark problems with up to 15 objectives. Empirical results fully demonstrate the superiority of our proposed method on all considered test instances. In addition, we extend this method to solve constrained problems having a large number of objectives. Compared to two other recently proposed constrained optimizers, our proposed method shows highly competitive performance on all the constrained optimization problems. [ABSTRACT FROM PUBLISHER]

Multiobjective evolutionary algorithm based on decomposition (MOEA/D), which bridges the traditional optimization techniques and population-based methods, has become an increasingly popular framework for evolutionary multiobjective optimization. It decomposes a multiobjective optimization problem (MOP) into a number of optimization subproblems. Each subproblem is handled by an agent in a collaborative manner. The selection of MOEA/D is a process of choosing solutions by agents. In particular, each agent has two requirements on its selected solution: one is the convergence toward the efficient front, the other is the distinction with the other agents’ choices. This paper suggests addressing these two requirements by defining mutual-preferences between subproblems and solutions. Afterwards, a simple yet effective method is proposed to build an interrelationship between subproblems and solutions, based on their mutual-preferences. At each generation, this interrelationship is used as a guideline to select the elite solutions to survive as the next parents. By considering the mutual-preferences between subproblems and solutions (i.e., the two requirements of each agent), the selection operator is able to balance the convergence and diversity of the search process. Comprehensive experiments are conducted on several MOP test instances with complicated Pareto sets. Empirical results demonstrate the effectiveness and competitiveness of our proposed algorithm. [ABSTRACT FROM PUBLISHER]

Multiobjective evolutionary algorithms have become prevalent and efficient approaches for solving multiobjective optimization problems. However, their performances deteriorate severely when handling many-objective optimization problems (MaOPs) due to the loss of selection pressure to drive the search toward the Pareto front and the ineffective design in diversity maintenance mechanism. This paper proposes a many-objective evolutionary algorithm (MaOEA) based on directional diversity (DD) and favorable convergence (FC). The main features are the enhancement of two selection schemes to facilitate both convergence and diversity. In the algorithm, a mating selection based on FC is applied to strengthen selection pressure while an environmental selection based on DD and FC is designed to balance diversity and convergence. The proposed algorithm is tested on 64 instances of 16 MaOPs with diverse characteristics and compared with seven state-of-the-art algorithms. Experimental results show that the proposed MaOEA performs competitively with respect to chosen state-of-the-art designs. [ABSTRACT FROM PUBLISHER]

Distribution system reconfiguration (DSR) is a multi-objective, nonlinear problem. This paper introduces a new, fast, nondominated sorting genetic algorithm (FNSGA) for the purpose of solving the DSR problem in normal operation by satisfying all objectives simultaneously with a relatively small number of generations and relatively short computation time. The objectives of the problem are to minimize real power losses and improve the voltage profile and load balancing index with minimum switching operations. Instead of generating several ranks from the nondominated set of solutions, this algorithm deals with only one rank; then the most suitable solution is chosen according to the operator's wishes. If there is no preference and all objectives have the same degree of importance, the best solution is determined by simply considering the sum of the normalized objective values. Also, a guided mutation operation is applied instead of a random one to speed up convergence. Radial system topology is satisfied using graph theory by formulating the branch-bus incidence matrix (BBIM) and checking the rank of each topology. To test the algorithm, it was applied to two widely studied test systems and a real one. The results show the efficiency of this algorithm as compared to other methods in terms of achieving all the goals simultaneously with reasonable population and generation sizes and without using a mutation rate, which is usually problem-dependent. [ABSTRACT FROM PUBLISHER]

Balancing convergence and diversity plays a key role in evolutionary multiobjective optimization (EMO). Most current EMO algorithms perform well on problems with two or three objectives, but encounter difficulties in their scalability to many-objective optimization. This paper proposes a grid-based evolutionary algorithm (GrEA) to solve many-objective optimization problems. Our aim is to exploit the potential of the grid-based approach to strengthen the selection pressure toward the optimal direction while maintaining an extensive and uniform distribution among solutions. To this end, two concepts—grid dominance and grid difference—are introduced to determine the mutual relationship of individuals in a grid environment. Three grid-based criteria, i.e., grid ranking, grid crowding distance, and grid coordinate point distance, are incorporated into the fitness of individuals to distinguish them in both the mating and environmental selection processes. Moreover, a fitness adjustment strategy is developed by adaptively punishing individuals based on the neighborhood and grid dominance relations in order to avoid partial overcrowding as well as guide the search toward different directions in the archive. Six state-of-the-art EMO algorithms are selected as the peer algorithms to validate GrEA. A series of extensive experiments is conducted on 52 instances of nine test problems taken from three test suites. The experimental results show the effectiveness and competitiveness of the proposed GrEA in balancing convergence and diversity. The solution set obtained by GrEA can achieve a better coverage of the Pareto front than that obtained by other algorithms on most of the tested problems. Additionally, a parametric study reveals interesting insights of the division parameter in a grid and also indicates useful values for problems with different characteristics. [ABSTRACT FROM AUTHOR]