Evolutionary Algorithm For Constrained Optimization Research Articles

A multi-population evolutionary algorithm based on knowledge transfer for constrained many-objective optimization

ABSTRACT Constrained Many-objective Optimization Problems (CMaOPs) are challenging in handling objectives and constraints simultaneously. Here, a novel Constrained Many-objective Optimization Evolutionary Algorithm (CMaOEA) based on Multi-population, Knowledge transfer and Improved environmental selection called CMaMKI is proposed to handle CMaOPs. The proposed framework evolves a task population to solve the original CMaOP and evolves another population to solve a helper problem derived from the original one. To assist solving the original CMaOP, a knowledge expression and transfer strategy is designed to share useful information in the helper population with the task population. Meanwhile, to balance population convergence, diversity and feasibility, an enhanced environmental selection strategy is devised by combining the ε-constrained technique, θ-dominance and subregional density evaluation. The proposed algorithm is evaluated and contrasted with six state-of-the-art algorithms on a set of benchmark CMaOPs. The experimental results demonstrate the superiority and competitiveness of the proposed method.

A novel tri-stage with reward-switching mechanism for constrained multiobjective optimization problems

The effective exploitation of infeasible solutions plays a crucial role in addressing constrained multiobjective optimization problems (CMOPs). However, existing constrained multiobjective optimization evolutionary algorithms (CMOEAs) encounter challenges in effectively balancing objective optimization and constraint satisfaction, particularly when tackling problems with complex infeasible regions. Subsequent to the prior exploration, this paper proposes a novel tri-stage with reward-switching mechanism framework (TSRSM), including the push, pull, and repush stages. Each stage consists of two coevolutionary populations, namely Pop1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Pop}}_1$$\\end{document} and Pop2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Pop}}_2$$\\end{document}. Throughout the three stages, Pop1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Pop}}_1$$\\end{document} is tasked with converging to the constrained Pareto front (CPF). However, Pop2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Pop}}_2$$\\end{document} is assigned with distinct tasks: (i) converging to the unconstrained Pareto front (UPF) in the push stage; (ii) utilizing constraint relaxation technique to discover the CPF in the pull stage; and (iii) revisiting the search for the UPF through knowledge transfer in the repush stage. Additionally, a novel reward-switching mechanism (RSM) is employed to transition between different stages, considering the extent of changes in the convergence and diversity of populations. Finally, the experimental results on three benchmark test sets and 30 real-world CMOPs demonstrate that TSRSM achieves competitive performance when compared with nine state-of-the-art CMOEAs. The source code is available at https://github.com/Qu-jq/TSRSM.

ATM-R: An Adaptive Tradeoff Model With Reference Points for Constrained Multiobjective Evolutionary Optimization.

The goal of constrained multiobjective evolutionary optimization is to obtain a set of well-converged and well-distributed feasible solutions. To achieve this goal, a delicate tradeoff must be struck among feasibility, diversity, and convergence. However, balancing these three elements simultaneously through a single tradeoff model is nontrivial, mainly because the significance of each element varies in different evolutionary phases. As an alternative approach, we adapt distinct tradeoff models in various phases and introduce a novel algorithm named adaptive tradeoff model with reference points (ATM-R). In the infeasible phase, ATM-R takes the tradeoff between diversity and feasibility into account, aiming to move the population toward feasible regions from diverse search directions. In the semi-feasible phase, ATM-R promotes the transition from "the tradeoff between feasibility and diversity" to "the tradeoff between diversity and convergence." This transition is instrumental in discovering an adequate number of feasible regions and accelerating the search for feasible Pareto optima in succession. In the feasible phase, ATM-R places an emphasis on balancing diversity and convergence to obtain a set of feasible solutions that are both well-converged and well-distributed. It is worth noting that the merits of reference points are leveraged in ATM-R to accomplish these tradeoff models. Also, in ATM-R, a multiphase mating selection strategy is developed to generate promising solutions beneficial to different evolutionary phases. Systemic experiments on a diverse set of benchmark test functions and real-world problems demonstrate that ATM-R is effective. When compared to eight state-of-the-art constrained multiobjective optimization evolutionary algorithms, ATM-R consistently demonstrates its competitive performance.

A constrained multi-objective optimization algorithm based on coordinated strategy of archive and weight vectors

When dealing with Constrained Multi-objective Optimization Problems (CMOPs) and struggling to enhance feasibility, convergence and diversity, the researchers of Constrained Multi-objective Optimization Evolutionary Algorithms (CMOEAs) gravitate toward feasibility or take precedence to preserve well-converged solutions ignoring diversity in the past. To compensate for the defects, the paper proposes CMOEA-MSWA to guide the search of infeasible regions by coordinated strategy of archive and weight vectors. Firstly, the archive carrying the information of population diversity updates the weight vectors. Secondly, the updated weight vectors perpetuate the diversity information to the search of infeasible regions. The circular effects between strategies promote the detection of infeasible solutions with good objectives and exhibit competitive performance in terms of spread and evenness. To testify the versatility the CMOEA-MSWA in enhancing diversity, the comprehensive performance is evaluated firstly and the diversity analysis of the CMOEA-MSWA is carried out on four benchmark suites with 34 test instances, where the number of objectives for some of test problems is scaled from three to five. In comparison with five state-of-the-art CMOEAs, the proposed algorithm yields highly competitive performance in diversity on different types of CMOPs. In addition, the effectiveness of collaboration between archive and weight vectors on handling infeasible solutions is also verified.

A Constrained Many-Objective Optimization Evolutionary Algorithm With Enhanced Mating and Environmental Selections.

Unlike the considerable research on solving many-objective optimization problems (MaOPs) with evolutionary algorithms (EAs), there has been much less research on constrained MaOPs (CMaOPs). Generally, to effectively solve CMaOPs, an algorithm needs to balance feasibility, convergence, and diversity simultaneously. It is essential for handling CMaOPs yet most of the existing research encounters difficulties. This article proposes a novel constrained many-objective optimization EA with enhanced mating and environmental selections, namely, CMME. It can be featured as: 1) two novel ranking strategies are proposed and used in the mating and environmental selections to enrich feasibility, diversity, and convergence; 2) a novel individual density estimation is designed, and the crowding distance is integrated to promote diversity; and 3) the \θ-dominance is used to strengthen the selection pressure on promoting both the convergence and diversity. The synergy of these components can achieve the goal of balancing feasibility, convergence, and diversity for solving CMaOPs. The proposed CMME is extensively evaluated on 13 CMaOPs and 3 real-world applications. Experimental results demonstrate the superiority and competitiveness of CMME over nine related algorithms.

A novel three-stage multi-population evolutionary algorithm for constrained multi-objective optimization problems

Lots of real-world optimization problems are inherently constrained multi-objective optimization problems (CMOPs), but the existing constrained multi-objective optimization evolutionary algorithms (CMOEAs) often fail to balance convergence and diversity effectively. Therefore, a novel constrained multi-objective optimization evolutionary algorithm based on three-stage multi-population coevolution (CMOEA-TMC) for complex CMOPs is proposed. CMOEA-TMC contains two populations, called mainPop and helpPop, which evolve with and without consideration of constraints, respectively. The proposed algorithm divides the search process into three stages. In the first stage, fast convergence is achieved by transforming the original multi-objective problems into multiple single-objective problems. Coarse-grained parallel evolution of subpopulations in mainPop and guidance information provided by helpPop can facilitate mainPop to quickly approach the Pareto front. In the second stage, the objective function of mainPop changes to the original problem. Coevolution of mainPop and helpPop by sharing offsprings can produce solutions with better diversity. In the third stage, the mining of the global optimal solutions is performed, discarding helpPop to save computational resources. For CMOEA-TMC, the combination of parallel evolution, coevolution, and staging strategy makes it easier for mainPop to converge and maintain good diversity. Experimental results on 33 benchmark CMOPs and a real-world boiler combustion optimization case show that CMOEA-TMC is more competitive than the other five advanced CMOEAs.

Differential evolution with an adaptive penalty coefficient mechanism and a search history exploitation mechanism

A critical issue of constrained optimization evolutionary algorithms (COEAs) is how to balance the minimization of the objective function and the reduction of the constraint violation during the optimization process. This paper proposes a differential evolution (DE) algorithm with an adaptive penalty coefficient mechanism and a search history exploitation mechanism (called APCSH) to address this issue. Specifically, APCSH includes a learning stage and an evolving stage. In the learning stage, the correlation between the constraint violation and objective function values is calculated. Then, in the evolving stage, the correlation is used with a sigmoid function to determine the upper and lower bounds of the penalty coefficient, followed by a ranking strategy to assign the penalty coefficient to each solution. Furthermore, an exponentially weighted average method is used to update the search direction of solutions to promote convergence. Additionally, the scaling factor and crossover rate of DE are tuned automatically based on the successful search history. Many numerical experiments are conducted on 18 CEC2010 benchmark problems, 28 CEC2017 benchmark problems, and 4 heat pipe constraint optimization problems to demonstrate the advantages of the proposed algorithm over some state-of-the-art COEAs.

A Surrogate-Assisted Constrained Optimization Evolutionary Algorithm By Searching Multiple Kinds of Global and Local Regions

A Surrogate-Assisted Constrained Optimization Evolutionary Algorithm By Searching Multiple Kinds of Global and Local Regions

Dynamic Constrained Boundary Method for Constrained Multi-Objective Optimization

When solving complex constrained problems, how to efficiently utilize promising infeasible solutions is an essential issue because these promising infeasible solutions can significantly improve the diversity of algorithms. However, most existing constrained multi-objective evolutionary algorithms (CMOEAs) do not fully exploit these promising infeasible solutions. In order to solve this problem, a constrained multi-objective optimization evolutionary algorithm based on the dynamic constraint boundary method is proposed (CDCBM). The proposed algorithm continuously searches for promising infeasible solutions between UPF (the unconstrained Pareto front) and CPF (the constrained Pareto front) during the evolution process by the dynamically changing auxiliary population of the constraint boundary, which continuously provides supplementary evolutionary directions to the main population and improves the convergence and diversity of the main population. Extensive experiments on three well-known test suites and three real-world constrained multi-objective optimization problems demonstrate that CDCBM is more competitive than seven state-of-the-art CMOEAs.

Evolutionary constrained optimization with hybrid constraint-handling technique

In constrained optimization evolutionary algorithms (COEAs), constraint-handling technique is used to balance the objective function and constraints, but how to achieve this balance is a very important problem. We found that the information of the population during the evolution process can reflect the current situation of the population, whether the population are inside the feasible region (feasible situation), or are near the boundary of the feasible region (semi-feasible situation), or far away from the feasible region (infeasible situation). Therefore, corresponding constraint-handling method are designed according to the information of each situation—Hybrid Constraint-handling Technique (HCT). The information of the population evolution process is used by HCT to maintain the objective function and constraints balance, and combines the evolutionary algorithm and HCT to propose ECO-HCT to solve COPs. Meanwhile, in infeasible situation, an elite replacement strategy is proposed to help the population accumulate experience. In addition, a criterion for judging that the population falls into the local optimum in the infeasible region and a simple restart mechanism are designed. They can help the population jump out of the local optimum in the infeasible region and effectively improve the algorithm’s ability to solve complex COPs. The 24 constraint test functions from IEEE CEC2006, the 28 constraint test functions from IEEE CEC2017, and three constrained engineering design problems are used to verify the effectiveness and efficiency of the proposed ECO-HCT. Experimental results show that ECO-HCT has very competitive performance compared with other advanced methods.

Evolutionary Algorithm with Dynamic Population Size for Constrained Multiobjective Optimization

The core task of constrained multiobjective optimization is to achieve a tradeoff between exploration and exploitation as well as a tradeoff between constraints and objectives. We present an effective evolutionary algorithm with a dynamic population size (DPSEA) to achieve these two tradeoffs. In order to balance exploration and exploitation, the population size of DPSEA decreases continually as generation increases. In this manner, a bigger population size can encourage exploration in the early stage, while a smaller one can promote exploitation in the later stage. Aiming to balance constraints and objectives, a two-stage environmental selection strategy is proposed. In this strategy, objective information and constraint information is used to select two sets of solutions, respectively. Note that the sizes of these two sets are also adjusted dynamically. In this way, the information of both objectives and constraints can be used. Moreover, a novel mating selection strategy is designed to select promising parents. By assembling the above processes, DPSEA is able to achieve the two tradeoffs which are critical to constrained multiobjective optimization. Experiments on three sets of benchmark test functions owning difficult characteristics validate that DPSEA is competitive against some state-of-the-art constrained multiobjective optimization evolutionary algorithms.

Elliptical Slot Microstrip Patch Antenna Design Based on a Dynamic Constrained Multiobjective Optimization Evolutionary Algorithm

Most evolutionary optimization algorithms have already been used for antenna design and shown promising results on improving the performance of the antenna. However, for many real-world antenna optimization problems, they are difficult to solve in that there are highly constrained and multimodal difficulty. These difficulties impede the development of antenna design. In this paper, an elliptical slot microstrip patch antenna design with these difficulties is modeled as a constrained optimization problem (COP). To address the problem, a Dynamic Constrained Multiobjective Optimization Evolutionary Algorithm(DCMOEA) is used. The experimental results show that the optimum antenna with satisfying the design requirement is obtained, and as well as we find the radiation patch should be a whole ellipse instead of subtracting with two ellipses.

Elliptical Slot Microstrip Patch Antenna Design Based on a Dynamic Constrained Multiobjective Optimization Evolutionary Algorithm

Most evolutionary optimization algorithms have already been used for antenna design and shown promising results on improving the performance of the antenna. However, for many real-world antenna optimization problems, they are difficult to solve in that there are highly constrained and multimodal difficulty. These difficulties impede the development of antenna design. In this paper, an elliptical slot microstrip patch antenna design with these difficulties is modeled as a constrained optimization problem (COP). To address the problem, a Dynamic Constrained Multiobjective Optimization Evolutionary Algorithm(DCMOEA) is used. The experimental results show that the optimum antenna with satisfying the design requirement is obtained, and as well as we find the radiation patch should be a whole ellipse instead of subtracting with two ellipses.

An adaptive fuzzy penalty method for constrained evolutionary optimization

Penalty function is well-known for constrained evolutionary optimization. An open question in the penalty function is how to tune the penalty coefficient. This paper proposes an adaptive fuzzy penalty method to address this issue, where the coefficient is adjusted at both the individual level and the population level. At the individual level, each individual chooses a penalty coefficient from a predefined domain according to some fuzzy rules. At the population level, the domain of the crisp output is adjusted adaptively by using population information. To enhance the population diversity, an effective mutation scheme is developed. Due to its numerous merits, differential evolution is used to design a search algorithm. By the above processes, a constrained optimization evolutionary algorithm called AFPDE is proposed. Since the objective function value and the degree of constraint violation are normalized, AFPDE is less problem-dependent than the seminal work of the fuzzy penalty method. AFPDE introduces a lower penalty value in the early stage of AFPDE while a higher one in the later stage. Thus, it can escape local optima in the infeasible region. Experiments on three well-known benchmark test sets and two mechanical design problems validate that AFPDE is competitive.

On the Cooperation Between Evolutionary Algorithms and Constraint Handling Techniques: A Further Empirical Study

Cooperation between Evolutionary Algorithms (EAs) and Constraint Handling Techniques (CHTs) plays an important role in Constrained Optimization Evolutionary Algorithms (COEAs). A constrained composite differential evolution (C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> oDE) sets a good example on the cooperation. This paper tries to further study the inner mechanism, i.e., the effect of different methods on generating and selecting solutions. In the solution generating part, a new method, which adopts stochastic ranking in selecting the best individual in the DE mutation operators, is proposed. In the solution selecting part, a new constraint handling technique combination is added. The experiments on benchmark functions from IEEE CEC2006 verify the effect. During the experiment, it is found that a new cooperation manner performs better than C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> oDE, which reflects the importance of cooperation.

Evolutionary dynamic constrained optimization: Test suite construction and algorithm comparisons

Many real-world applications can be modelled as dynamic constrained optimization problems (DCOPs). Due to the fact that objective function and/or constraints change over time, solving DCOPs is a challenging task. Although solving DCOPs by evolutionary algorithms has attracted increasing interest in the community of evolutionary computation, the design of benchmark test functions of DCOPs is still insufficient. Therefore, we propose a test suite for DCOPs. A dynamic unconstrained optimization benchmark with good time-varying characteristics, called moving peaks benchmark, is chosen to be the objective function of our test suite. In addition, we design adjustable dynamic constraints, by which the size, number, and change severity of the feasible regions can be flexibly controlled. Furthermore, the performance of three dynamic constrained optimization evolutionary algorithms is tested on the proposed test suite, one of which is presented in this paper, named dynamic constrained optimization differential evolution (DyCODE). DyCODE includes three main phases: 1) the first phase intends to enter the feasible region from different directions promptly via a multi-population search strategy; 2) in the second phase, some excellent individuals chosen from the first phase form a new population to search for the optimal solution of the current environment; and 3) the third phase combines the memory individuals of the first two phases with some randomly generated individuals to re-initialize the population for the next environment. From the experiments, one can understand the strengths and weaknesses of the three compared algorithms for solving DCOPs in depth. Moreover, we also give some suggestions for researchers to apply these three algorithms on different occasions.

Utilizing the Correlation Between Constraints and Objective Function for Constrained Evolutionary Optimization

When solving constrained optimization problems by evolutionary algorithms, the core issue is to balance constraints and objective function. This paper is the first attempt to utilize the correlation between constraints and objective function to keep this balance. First of all, the correlation between constraints and objective function is mined and represented by a correlation index. Afterward, a weighted sum updating approach and an archiving and replacement mechanism are proposed to make use of this correlation index to guide the evolution. By the above process, a novel constrained optimization evolutionary algorithm is presented. Experiments on a broad range of benchmark test functions indicate that the proposed method shows better or at least competitive performance against other state-of-the-art methods. Moreover, the proposed method is applied to the gait optimization of humanoid robots.

An improved constrained multi-objective optimization evolutionary algorithm for carbon fibre drawing process

ABSTRACTIn this paper, an improved ϵCMOEA/D-DE is proposed to improve the performance of CMOP algorithm and achieve parameter optimization in carbon fibre drawing process. In order to avoid overusing the infeasible solutions, two repair operators are introduced in the population evolution model. More specially, during the evolutionary process, when the constraint violation of infeasible solution exceeds a tolerance threshold, the proposed two repair operators are used to find a better solution to repair the infeasible solution. On the other hand, in order to enhance the convergence rate of the Differential Evolution (DE), a modified DE is proposed. Then, an ϵCMOEA/D-mDE-RO is proposed by incorporating these two improved strategies into the ϵCMOEA/D-DE. Subsequently, the performance of the proposed ϵCMOEA/D-mDE-RO is evaluated on the constrained test problem series and the experiment shows that the proposed algorithm outperforms the existing ϵCMOEA/D-DE. Finally, in order to further illustrate the application potential, the proposed algorithm is successfully applied in optimizing carbon fibre drawing process and the optimal draw ratio vector is obtained.

An improved teaching-learning-based optimization for constrained evolutionary optimization

When extending a global optimization technique for constrained optimization, we must balance not only diversity and convergence but also constraints and objective function. Based on these two criteria, the famous teaching-learning-based optimization (TLBO) is improved for constrained optimization. To balance diversity and convergence, an efficient subpopulation based teacher phase is designed to enhance diversity, while a ranking-differential-vector-based learner phase is proposed to promote convergence. In addition, how to select the teacher in the teacher phase and how to rank two solutions in the learner phase have a significant impact on the tradeoff between constraints and objective function. To address this issue, a dynamic weighted sum is formulated. Furthermore, a simple yet effective restart strategy is proposed to settle complicated constraints. By adopting the ε constraint-handling technique as the constraint-handling technique, a constrained optimization evolutionary algorithm, i.e., improved TLBO (ITLBO), is proposed. Experiments on a broad range of benchmark test functions reveal that ITLBO shows better or at least competitive performance against other constrained TLBOs and some other constrained optimization evolutionary algorithms.

Differential evolution with adaptive trial vector generation strategy and cluster-replacement-based feasibility rule for constrained optimization

Constrained optimization problems (COPs) are common in many fields. To solve such problems effectively, in this paper, we propose a new constrained optimization evolutionary algorithm (COEA) named CACDE that combines an adaptive trial vector generation strategy-based differential evolution (DE) algorithm with a cluster-replacement-based feasibility rule. In CACDE, some potential mutation strategies, scale factors and crossover rates are stored in candidate pools, and each element in the pools is assigned a selection probability. During the trial vector generation stage, the mutation strategy, scale factor and crossover rate for each target vector are competitively determined based on these selection probabilities. Meanwhile, the selection probabilities are dynamically updated based on statistical information learned from previous searches in generating improved solutions. Moreover, to alleviate the greediness of the feasibility rule, the main population is divided into several clusters, and one vector in each cluster is conditionally replaced with an archived infeasible vector with a low objective value. The superior performance of CACDE is validated via comparisons with some state-of-the-art COEAs over 2 sets of artificial problems and 5 widely used mechanical design problems. The results show that CACDE is an effective approach for solving COPs, basically due to the use of adaptive DE and cluster-replacement-based feasibility rule.