Constrained multiobjective optimization problems often have complex feasible regions and constrained Pareto fronts. These factors bring great challenges to current constrained multiobjective ...optimization evolutionary algorithms (CMOEAs). To solve this problem and further balance the objective optimization and constraint satisfaction, we propose an indexesbased and partial restart-based constrained multiobjective optimization algorithm (IRCMO). In IRCMO, a two-stage (i.e., development and enhancement) and tri-population framework is designed. IRCMO adopts the aggregative indexes-based evaluation and adaptive collaborative partial restart strategy to assist the evolution of the first and second populations. The third population is obtained by directed sampling, which is mostly located at the boundary of the feasible region and enhances the exploration ability of extreme solutions. At the end of each generation, a progressive dual-archive strategy is designed to screen the solutions distributed uniformly from three populations. Experimental results demonstrate that IRCMO is superior to the other six state-of-the-art CMOEAs on several constraint benchmark suites and real-world problems.
Constrained multiobjective optimization problems (CMOPs) are challenging because of the difficulty in handling both multiple objectives and constraints. While some evolutionary algorithms have ...demonstrated high performance on most CMOPs, they exhibit bad convergence or diversity performance on CMOPs with small feasible regions. To remedy this issue, this article proposes a coevolutionary framework for constrained multiobjective optimization, which solves a complex CMOP assisted by a simple helper problem. The proposed framework evolves one population to solve the original CMOP and evolves another population to solve a helper problem derived from the original one. While the two populations are evolved by the same optimizer separately, the assistance in solving the original CMOP is achieved by sharing useful information between the two populations. In the experiments, the proposed framework is compared to several state-of-the-art algorithms tailored for CMOPs. High competitiveness of the proposed framework is demonstrated by applying it to 47 benchmark CMOPs and the vehicle routing problem with time windows.
Constrained multiobjective optimization problems (CMOPs) are frequently encountered in real-world applications, which usually involve constraints in both the decision and objective spaces. However, ...current artificial CMOPs never consider constraints in the decision space (i.e., decision constraints) and constraints in the objective space (i.e., objective constraints) at the same time. As a result, they have a limited capability to simulate practical scenes. To remedy this issue, a set of CMOPs, named DOC, is constructed in this paper. It is the first attempt to consider both the decision and objective constraints simultaneously in the design of artificial CMOPs. Specifically, in DOC, various decision constraints (e.g., inequality constraints, equality constraints, linear constraints, and nonlinear constraints) are collected from real-world applications, thus making the feasible region in the decision space have different properties (e.g., nonlinear, extremely small, and multimodal). On the other hand, some simple and controllable objective constraints are devised to reduce the feasible region in the objective space and to make the Pareto front have diverse characteristics (e.g., continuous, discrete, mixed, and degenerate). As a whole, DOC poses a great challenge for a constrained multiobjective evolutionary algorithm (CMOEA) to obtain a set of well-distributed and well-converged feasible solutions. In order to enhance current CMOEAs' performance on DOC, a simple and efficient two-phase framework, named ToP, is proposed in this paper. In ToP, the first phase is implemented to find the promising feasible area by transforming a CMOP into a constrained single-objective optimization problem. Then in the second phase, a specific CMOEA is executed to obtain the final solutions. ToP is applied to four state-of-the-art CMOEAs, and the experimental results suggest that it is quite effective.
Handling constrained multiobjective optimization problems (CMOPs) is extremely challenging, since multiple conflicting objectives subject to various constraints require to be simultaneously ...optimized. To deal with CMOPs, numerous constrained multiobjective evolutionary algorithms (CMOEAs) have been proposed in recent years, and they have achieved promising performance. However, there has been few literature on the systematic review of the related studies currently. This article provides a comprehensive survey for evolutionary constrained multiobjective optimization. We first review a large number of CMOEAs through categorization and analyze their advantages and drawbacks in each category. Then, we summarize the benchmark test problems and investigate the performance of different constraint handling techniques (CHTs) and different algorithms, followed by some emerging and representative applications of CMOEAs. Finally, we discuss some new challenges and point out some directions of the future research in the field of evolutionary constrained multiobjective optimization.
For solving constrained multiobjective optimization problems (CMOPs), many algorithms have been proposed in the evolutionary computation research community for the past two decades. Generally, the ...effectiveness of an algorithm for CMOPs is evaluated by artificial test problems. However, after a brief review of current artificial test problems, we have found that they are not well-designed and fail to reflect the characteristics of real-world applications (e.g., small feasibility ratio). Thus, in this paper, we first propose a new constraint construction method to facilitate the systematic design of test problems. Then, on the basis of this method, we design a new test suite consisting of 14 instances, which covers diverse characteristics extracted from real-world CMOPs and can be divided into four types. Considering that the comprehensive performance comparisons among the constraint-handling techniques (CHTs) remain scarce, we choose several representative CHTs and compare their performance on our test suite. The performance comparisons identify the strengths and weaknesses of different CHTs on different types of CMOPs and provide guidelines on how to select/design a CHT in a specific scenario.
The main challenge in constrained multiobjective optimization problems (CMOPs) is to appropriately balance convergence, diversity and feasibility. Their imbalance can easily cause the failure of a ...constrained multiobjective evolutionary algorithm (CMOEA) in converging to the Pareto-optimal front with diverse feasible solutions. To address this challenge, we propose a dual-population-based evolutionary algorithm, named c-DPEA, for CMOPs. c-DPEA is a cooperative coevolutionary algorithm which maintains two collaborative and complementary populations, termed Population1 and Population2 . In c-DPEA, a novel self-adaptive penalty function, termed saPF , is designed to preserve competitive infeasible solutions in Population1 . On the other hand, infeasible solutions in Population2 are handled using a feasibility-oriented approach. To maintain an appropriate balance between convergence and diversity in c-DPEA, a new adaptive fitness function, named bCAD , is developed. Extensive experiments on three popular test suites comprehensively validate the design components of c-DPEA. Comparison against six state-of-the-art CMOEAs demonstrates that c-DPEA is significantly superior or comparable to the contender algorithms on most of the test problems.
Both objective optimization and constraint satisfaction are crucial for solving constrained multiobjective optimization problems, but the existing evolutionary algorithms encounter difficulties in ...striking a good balance between them when tackling complex feasible regions. To address this issue, this article proposes a two-stage evolutionary algorithm, which adjusts the fitness evaluation strategies during the evolutionary process to adaptively balance objective optimization and constraint satisfaction. The proposed algorithm can switch between the two stages according to the status of the current population, enabling the population to cross the infeasible region and reach the feasible regions in one stage, and to spread along the feasible boundaries in the other stage. Experimental studies on four benchmark suites and three real-world applications demonstrate the superiority of the proposed algorithm over the state-of-the-art algorithms, especially on problems with complex feasible regions.
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.
Constrained multi-objective optimization problems (CMOPs) contain the satisfaction of various constraints and optimization of multiple objectives simultaneously, thus they are extremely challenging. ...Although many constrained multi-objective evolutionary algorithms (CMOEAs) have been proposed, they ignore the information of each constraint, which might help utilize more various infeasible solutions to improve the search ability of the population. Therefore, this paper proposes a new dual-population CMOEA to solve CMOPs, in which a dynamic constraint processing mechanism and a dynamic resource allocating scheme are designed. To be specific, the proposed algorithm evolves two populations, which adopt different mechanisms to handle constraints respectively. The main population directly optimizes all constraints to find the feasible Pareto optimal solutions, which can improve the feasibility. The auxiliary population adopts a dynamic constraint processing mechanism, which gradually increases the number of constraints being processed, so as to fully utilize various infeasible solutions to help find feasible regions. Moreover, a new dynamic resource allocating scheme is proposed to reasonably allocate the limited computational resources to the two populations according to their performance feedback. Experimental results on three test suites and ten practical problems show that the proposed algorithm has a better or competitive performance compared with several state-of-the-art CMOEAs.
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