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

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

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

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

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