A multi-stage evolutionary algorithm for multi-objective optimization with complex constraints

Abstract

Constrained multi-objective optimization problems (CMOPs) are difficult to handle because objectives and constraints need to be considered simultaneously, especially when the constraints are extremely complex. Some recent algorithms work well when dealing with CMOPs with a simple feasible region; however, the effectiveness of most algorithms degrades considerably for CMOPs with complex feasible regions. To address this issue, this paper proposes a multi-stage evolutionary algorithm, where constraints are added one after the other and handled in different stages of evolution. Specifically, in the early stages, the algorithm only considers a small number of constraints, which can make the population efficiently converge to the potential feasible region with good diversity. As the algorithm moves to the later stages, more constraints are considered to search the optimal solutions based on the solutions obtained in the previous stages. Furthermore, a strategy for sorting the constraint-handling priority according to the impact on the unconstrained Pareto front is proposed, which can accelerate the convergence of the algorithm. Experimental results on five benchmark suites and three real-world applications showed that the proposed algorithm outperforms several state-of-the-art constraint multi-objective evolutionary algorithms when dealing with CMOPs, especially for problems 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 as

$$\begin{array}{cc}\mathrm{Minimize}\hfill & \mathbf{F}(\mathbf{x})=(f_1(\mathbf{x}),\cdots ,f_m(\mathbf{x}))\hfill \\ \text{subject to}\hfill & \mathbf{x}\in \mathrm{\Omega }\hfill \\ \hfill & g_i(\mathbf{x})⩽0,i=1,\cdots ,p\hfill \\ \hfill & h_j(\mathbf{x})=0,j=1,\cdots ,q\hfill \end{array}$$

where $\mathbf{x}=(x_1,\cdots ,x_d)\in \mathrm{\Omega }$ is a solution consisting of d decision variables, $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ is the decision space, $\mathbf{F}:\mathrm{\Omega }\to {\mathbb{R}}^m$ consists of m objectives, $g_i(\mathbf{x})⩽0$ is the i-th inequality constraint, and $h_j(\mathbf{x})=0$ is the j-th equality constraint. The overall constraint violation degree of a solution $\mathbf{x}$ adopted in this paper is defined as

$$C(\mathbf{x})={\displaystyle \sum _{k=1}^p}\max \left\{0,g_k(\mathbf{x})\right\}+{\displaystyle \sum _{\ell =1}^q}\left|h_{\ell }(\mathbf{x})\right|$$

A solution $\mathbf{x}$ is feasible if $C(\mathbf{x})=0$; 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:

• 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.

• 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.

• 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.

The remainder of this paper is organized as follows. In Section 2, we first introduce existing CMOEAs with different constraint-handling techniques for CMOPs, and then we discuss the motivation for this work. In Section 3, the proposed algorithm is explained in detail. Experimental details on the benchmark CMOPs and real-world applications are given in Section 4. Finally, we present the conclusions and future work in Section 5.