A multi-stage evolutionary algorithm for multi-objective optimization with complex constraints
Introduction
Constrained multi-objective optimization problems (CMOPs) widely exist in many real-world applications, such as optimal software product selection [44], knapsack problems [3], and capacitated arc routing problems [1]. All these problems need to optimize some conflicting objectives and meet a set of constraints simultaneously. Without loss of generality, a CMOP can be defined aswhere is a solution consisting of d decision variables, is the decision space, consists of m objectives, is the i-th inequality constraint, and is the j-th equality constraint. The overall constraint violation degree of a solution adopted in this paper is defined asA solution is feasible if ; otherwise, it is infeasible. The feasible region is composed of all solutions that satisfy all the constraints.
Over the last two decades, a variety of promising evolutionary algorithms have been developed, especially for solving multi-objective optimization problems [6]. These multi-objective evolutionary algorithms (MOEAs) can be roughly divided into four categories, namely evolutionary algorithms based on Pareto dominance, such as NSGA-II [11] and SPEA2 [50]; evolutionary algorithms based on decomposition, such as MOEA/D [46] and MOEA/D-AWA [34]; indicator-based evolutionary algorithms, such as IBEA [49] and AR-MOEA [36]; and mixed evolutionary algorithms, such as SRA [24] and Two_Arch [40]. Based on these four categories, many algorithms have been developed to solve various multi-objective optimization problems, i.e., large-scale multi-objective optimization problems [4], [17] and many-objective optimization problems [10], [36]. In recent years, constrained multi-objective optimization has attracted wide attention from researchers. An increasing number of MOEAs have been developed to solve such problems (MOEAs for solving CMOPs are thus named CMOEAs), such as NSGA-II-CDP [11] based on the constrained dominance principle, C-TAEA [25] based on the two-archive strategy, and ToP [27] based on the two-stage search process. Although these MOEAs achieve competitive performance in solving CMOPs, they suffer from considerable performance deterioration on some CMOPs [27], [12], where the constrained landscape is significantly complex, such as the discrete feasible region with a huge infeasible barrier (e.g., LIR-CMOP10 [12]) and only several narrow feasible regions (e.g., MW11 [28]), as shown in Fig. 1. Generally, these complex constrained landscapes are composed of more than one constraint. The CMOPs will become simpler and easier to be handled if we consider these constraints one by one and handle them in different stages of evolution since the constrained landscape with a small number of constraints will not be so complicated.
Following this idea, this paper proposes a multi-stage evolutionary algorithm for solving CMOPs. In contrast to most existing algorithms that regard all constraints as a whole and deal with them together, the proposed algorithm divides the constraint-handling process into multiple stages and deals with the constraints one stage by stage. The main contributions of this work are summarized as follows:
- 1)
A multi-stage CMOEA (MSCMO) is proposed for solving CMOPs with complex constraints. In the proposed algorithm, the constraints are added one by one and handled in different stages of evolution. In the early stages, the proposed algorithm only deals with a small number of constraints, which can make the population efficiently converge to the potential feasible region with good diversity. With each stage, more constraints are considered and can be handled more easily based on the solutions obtained in the previous stages.
- 2)
A strategy for sorting the constraint-handling priority according to the impact on the unconstrained Pareto front is suggested in the proposed MSCMO, which is used to determine the constraints to be handled at each stage. Experimental results on benchmark CMOPs showed that the proposed constraint-handling priority can accelerate the convergence of the algorithm.
- 3)
The results of a series of experiments on benchmark CMOPs and real-world applications show that our algorithm is very competitive in comparison with some state-of-the-art CMOEAs, especially on the CMOPs with complex constraints.
Section snippets
Existing CMOEAs
In general, existing CMOEAs can be classified into four categories according to the constraint-handling technology that they adopt.
The first category is the penalty function approach. The main idea is to construct a penalty term based on the degree of individual constraint violation. A penalty fitness function is first constructed by adding a penalty term to the objective function . Then, is used to evaluate the individuals. The penalty function method can be
Overview of the proposed MSCMO
Algorithm 1. Procedure of the proposed MSCMO Algorithm 2, Constraint_Priority_Determinate
The general flow of the proposed MSCMO is presented in Fig. 8. First, the constraint-handling priority is determined, which is used to decide the constraints to be handled at each stage. Then, the algorithm initializes a population P with n individuals randomly, which is evolved generation by generation until the maximum evaluations are reached. At each generation, the algorithm repeats the following
Experimental study
To verify the performance of the proposed algorithm, we performed a series of experiments, the results of which are discussed in this section. First, the proposed MSCMO was compared with five state-of-the-art CMOEAs on five benchmark test suites. Then, we verified the effectiveness of the constraint-handling priority strategy suggested in the proposed MSCMO. Finally, to further verify the performance of the proposed MSCMO, we tested several CMOPs from real-world applications. All our
Conclusions
In this paper, we proposed a multi-stage CMOEA for solving CMOPs with a relatively complex feasible region. Specifically, in the proposed algorithm, the constraints are added one by one and handled in different stages of evolution. In the early stages, only a small number of constraints are considered, which makes the population efficiently converge to the potential feasible region with good diversity. As the proposed algorithm enters the next stage, more constraints are considered, and the
CRediT authorship contribution statement
Haiping Ma: Investigation, Methodology, Writing - original draft. Haoyu Wei: Data curation, Methodology, Visualization. Ye Tian: Writing - review & editing. Ran Cheng: Writing - review & editing. Xingyi Zhang: Conceptualization, Investigation, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Key Research and Development Project (2018AAA0100105), the National Natural Science Foundation of China (61822301, 61672033, 61906001, and U1804262), the Hong Kong Scholars Program (XJ2019035), the Anhui Provincial Natural Science Foundation (1808085J06 and 1908085QF271), and the CCF-Tencent Open Research Fund (CCF-Tencent RAGR20200121).
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