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.
Various real-world problems can be attributed to constrained multiobjective optimization problems (CMOPs). Although there are various solution methods, it is still very challenging to automatically ...select efficient solving strategies for CMOPs. Given this, a process knowledge-guided constrained multiobjective autonomous evolutionary optimization method is proposed. First, the effects of different solving strategies on population states are evaluated in the early evolutionary stage. Then, the mapping model of population states and solving strategies is established. Finally, the model recommends subsequent solving strategies based on the current population state. This method can be embedded into existing evolutionary algorithms, which can improve their performances to different degrees. The proposed method is applied to 41 benchmarks and 30 dispatch optimization problems of the integrated coal mine energy system. Experimental results verify the effectiveness and superiority of the proposed method in solving CMOPs.
Solving multiobjective optimization problems (MOPs) through metaheuristic methods gets considerable attention. Based on the classical variation operators, several enhanced operators, as well as ...multiobjective optimization evolutionary algorithms, have been developed. Among these operators, the competitive swarm optimizer (CSO) exhibits promising performance. However, it encounters difficulties when tackling constrained MOPs (CMOPs) with large objective spaces or complex infeasible regions. In this article, a competitive and cooperative swarm optimizer is proposed, which contains two particle update strategies: 1) the CSO provides faster convergence speed to accelerate the approximation of the Pareto front and 2) the cooperative swarm optimizer suggests a mutual-learning strategy to enhance the ability to jump out of local feasible regions or local optima. We also present a new algorithm for CMOPs. The results on four benchmark suites with 47 instances demonstrate the superiority of our approach compared with other state-of-the-art methods. Additionally, its effectiveness on large-scale CMOPs has also been verified.
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.
Dynamic constrained multiobjective optimization problems (DCMOPs) abound in real-world applications and gain increasing attention in the evolutionary computation community. To evaluate the capability ...of an algorithm in solving DCMOPs, artificial test problems play a fundamental role. Nevertheless, some characteristics of real-world scenarios are not fully considered in the previous test suites, such as time-varying size, location and shape of feasible regions, the controllable change severity, as well as small feasible regions. Therefore, we develop the generators of objective functions and constraints to facilitate the systematic design of DCMOPs, and then a novel test suite consisting of nine benchmarks, termed as DCP, is put forward. To solve these problems, a dynamic constrained multiobjective evolutionary algorithm with a two-stage diversity compensation strategy (TDCEA) is proposed. Some initial individuals are randomly generated to replace historical ones in the first stage, improving the global diversity. In the second stage, the increment between center points of Pareto sets in the past two environments is calculated and employed to adaptively disturb solutions, forming an initial population with good diversity for the new environment. Intensive experiments show that the proposed test problems enable a good understanding of strengths and weaknesses of algorithms, and TDCEA outperforms other state-of-the-art comparative ones, achieving promising performance in tackling DCMOPs.
When solving constrained multiobjective optimization problems by evolutionary algorithm, the key challenge is how to achieve the balance among convergence, diversity, and feasibility. To deal with ...this challenge, a purpose-directed two-phase multiobjective differential evolution (PDTP-MDE) algorithm is developed in this paper. The main idea of PDTP-MDE is that the whole evolution process is divided into two sequential phases according to the expected purpose of each stage. To be specific, the first phase aims at keeping the balance between convergence and diversity, while the feasibility is taken as an auxiliary indicator. In this way, the population is capable of exploring different potential areas and avoiding to be trapped into local ones, thus providing more information about convergence and diversity for the later evolution process. Afterwards, the second phase mainly tends to maintain feasibility and diversity by selecting and using some promising infeasible solutions according to the population evolution status. In addition, an archive is maintained after each phase to preserve the superior feasible Pareto solutions found so far. By the above processes, the feasible Pareto front with well convergence and well diversity is obtained. The comprehensive experiments on 42 benchmark problems from three test suites demonstrate the superiority and competitiveness of the proposed PDTP-MDE, in comparison with other state-of-the-art constrained multiobjective evolutionary algorithms.
This paper proposes a new parameterless constraint-handling technique, named constrained probabilistic Pareto dominance (CPPD), for expensive constrained multiobjective optimization problems (CMOPs). ...In CPPD, when comparing two solutions, in terms of each original objective, we design a new objective for each solution, which is the negative product of two probabilities calculated based on the predicted fitness mean values and the uncertainty information provided by Kriging models: 1) the probability that this solution satisfies all constraints, denoted as PoF, and 2) the probability that this solution is better than the other on the original objective, denoted as PoB. It is evident that for each solution, PoF and PoB indicate its feasibility and its optimality on the corresponding original objective, respectively. Then, Pareto dominance based on new objectives is executed. As a result, both competitive feasible solutions and promising infeasible solutions with good diversity can be preserved by CPPD. These two kinds of solutions can help the population to exploit the located feasible parts and to explore new feasible parts, respectively. Further, based on CPPD, we develop a Pareto-based Kriging-assisted constrained multiobjective evolutionary algorithm (called PEA) to deal with expensive CMOPs with two or three objectives. Finally, PEA is generalized to solve expensive constrained many-objective optimization problems, named PEA+. The effectiveness of CPPD, PEA, and PEA+ is verified by comprehensive experiments.
Constrained multiobjective optimization has gained much interest in the past few years. However, constrained multiobjective optimization problems (CMOPs) are still unsatisfactorily understood. ...Consequently, the choice of adequate CMOPs for benchmarking is difficult and lacks a formal background. This paper takes a step towards addressing this issue by exploring CMOPs from a performance space perspective. First, it presents a novel performance assessment approach designed explicitly for constrained multiobjective optimization. This methodology offers a first attempt at simultaneously measuring the performance in approximating the Pareto front and constraint satisfaction. Secondly, it proposes an approach to measure the capability of the given optimization problem to differentiate among algorithm performances. Finally, this approach is used to compare eight frequently used artificial test suites of CMOPs. The experimental results reveal which suites are more efficient in discerning between four well-known multiobjective optimization algorithms.
Dynamic constrained multi-objective optimization problems (DCMOPs) involve objectives, constraints, and parameters that change over time. This kind of problem presents a greater challenge for ...evolutionary algorithms because it requires the population to quickly track the changing pareto-optimal set (PS) under constrained conditions while maintaining the feasibility and good distribution of the population. To address these challenges, this paper proposes a dynamic constrained multi-objective optimization algorithm based on co-evolution and diversity enhancement (CEDE), in which we have made improvements to both the static optimization and dynamic response parts, innovatively utilizing the valuable information latent in the optimization process to help the population evolve more comprehensively. The static optimization involves the co-evolution of three populations, through which their mutual synergy can more comprehensively identify potential true PS and provide more useful historical information for dynamic response. Additionally, to prevent the elimination of potentially valuable infeasible individuals (i.e., individuals that are not dominated by feasible individuals) due to pareto domination, we employ an archive set to store and update these individuals. When the environment changes, to effectively enhance population diversity under complex dynamic constraints and help the population to respond quickly to changes, we propose a diversity enhancement strategy, which includes a diversity maintenance strategy and a center point-based exploration strategy. This strategy effectively enhances population diversity in complex and changing environments, helping the population respond quickly to changes. The effectiveness of the algorithm is validated through two test sets. The experimental results show that CEDE can effectively use valuable information to cope with complex dynamic constraint environments. Compared with several of the most advanced algorithms, it is superior in 94% of the test problems, demonstrating strong competitiveness in handling DCMOPs.
