Surrogate-assisted MOEA/D for expensive constrained multi-objective optimization

Highlights

• A RBF-assisted MOEA/D framework is designed to achieve adaptive search.

• Different optimization states of subproblems are effectively identified by OSDM.

• Optimization state-driven search strategies are used for targeted suboptimization.

• The RBF accuracies are gradually improved by emphasizing specific points.

• The performance difference of two classical decompositions has been well studied.

Abstract

In this paper, an adaptive surrogate-assisted MOEA/D framework (ASA-MOEA/D) is proposed for solving computationally expensive constrained multi-objective optimization problems, in which three specific search strategies are adaptively implemented based on the optimization states of subproblems to achieve targeted searches for different subproblems. To maintain feasibility, the RBF-based local search models are constructed by comprehensively considering the orthogonal distance difference and constraint satisfaction information for guiding infeasible solutions of the infeasible subproblems into feasible regions. To maintain convergence, the RBF surrogates of the aggregated objective and constraints are employed to construct local search models for locating better feasible solutions. To maintain diversity, the subregions of unexplored subproblems are effectively explored by utilizing the valuable information of their neighboring elite solutions. Moreover, the solution with the maximum overall uncertainty of RBF surrogates is selected for progressively increasing the prediction accuracies of surrogates. Therefore, ASA-MOEA/D strikes an adaptive balance among diversity, feasibility and convergence with the assistance of RBF surrogates as the optimization progresses. Empirical studies on three classical test suites demonstrate that ASA-MOEA/D with tchebycheff approach achieves highly competitive performance over other four state-of-the-art algorithms.

Introduction

Constrained multi-objective optimization problems (CMOPs) involve the optimization of two or three objectives. In general, the CMOPs considered in this paper can be described as follows:

$$\begin{array}{c}\min \mathit{F}\left(\mathit{x}\right)={\left(f_1\left(\mathit{x}\right),\dots ,f_m\left(\mathit{x}\right)\right)}^T\\ s.t.\left\{\begin{array}{c}{g}_i\left(\mathit{x}\right)⩽0,i=1,\dots ,q\hfill \\ h_j\left(\mathit{x}\right)=0,j=1,\dots ,p\hfill \\ \mathit{x}\in \mathrm{\Omega }\hfill \end{array}\right.\end{array}$$

where $\mathit{x}={\left(x_1,\dots ,x_n\right)}^T$ is an n-dimensional decision vector, and $\mathrm{\Omega }={\left[x_i^L,x_i^U\right]}^n\subseteq {\mathfrak{R}}^n$ defines the decision space. $\mathit{F}:\mathrm{\Omega }\to {\mathfrak{R}}^m$ constitutes m conflicting objective functions, and $f_i\left(\mathit{x}\right)$ is the ith objective. $g_i\left(\mathit{x}\right)⩽0$ is the ith inequality constraint, and $h_j\left(\mathit{x}\right)=0$ is the jth equality constraint. The constraint violation (CV) of the jth constraint can be calculated as [1]:

$$C_j\left(\mathit{x}\right)=\left\{\begin{array}{c}\max (g_j\left(\mathit{x}\right),0),j=1,\dots ,q\hfill \\ \max (\left|h_j\left(\mathit{x}\right)\right|-\delta ,0),j=q+1,\dots ,p+q\hfill \end{array}\right.$$

where $\delta$ is a sufficiently small tolerance term (e.g., $\delta ={10}^{-6}$) used to relax equality constraints. Then $\mathit{x}$ is a feasible solution if the total constraint violation, i.e.,

$$C\left(\mathit{x}\right)=\sum _{j=1}^{p+q}{C}_j\left(\mathit{x}\right)$$

is equal to 0; otherwise, $\mathit{x}$ is an infeasible solution. Given two feasible solutions ${\mathit{x}}^1,{\mathit{x}}^2\in \mathrm{\Omega }$, it can be said that ${\mathit{x}}^1$ Pareto dominates ${\mathit{x}}^2$ (donated as ${\mathit{x}}^1\prec {\mathit{x}}^2$) in case $\mathit{F}\left({\mathit{x}}^1\right)$ is not worse than $\mathit{F}\left({\mathit{x}}^2\right)$ in any individual objective and it at least has one better objective. ${\mathit{x}}^{*}$ is a Pareto optimal solution when there does not exist any solution that Pareto dominates it. The set of all the Pareto optimal solutions is called the Pareto set (PS). Accordingly, The Pareto front (PF) is the image of the PS in the objective space.

Until now, evolutionary algorithms (EAs) [2], [3], [4] have shown excellent search abilities in solving many complex optimization problems. Generally, striking a well balance among convergence, diversity and feasibility is very critical when solving CMOPs. The multi-objective evolutionary algorithms (MOEAs) [5], [6], [7] have recently exhibited excellent search abilities on solving unconstrained multi-objective optimization problems for balancing convergence and diversity, and the constraint handling techniques (CHTs) [8], [9], [10] are capable of shifting the search of MOEAs into feasible regions of CMOPs for maintaining feasibility during the whole optimization process. Therefore, the effective combination between MOEAs and CHTs is able to achieve the balance among these three indicators. Generally speaking, MOEAs can be divided into three categories [11], [12], [13], [14], [15]. The first one is dominance-based MOEAs where different solutions are compared according to Pareto domination, such as non-dominated sorting and crowding distance ranking in NSGA-II [16], and Fuzzy Pareto dominance in FD-NSGA-II [17]. The second category is decomposition-based MOEAs, in which the multi-objective optimization problem is decomposed into a number of single or multi-objective optimization subproblems that are optimized simultaneously, including multi-objective evolutionary algorithm based on decomposition (MOEA/D) [5], reference vector guided-EA (RVEA) [18], angle-based adaptive penalty scheme in MOEA/D-AAP [19], and so on. The third category is indicator-based MOEAs such as hypervolume estimation algorithm for multi-objective optimization in HyPE [20], s-metric selection evolutionary multi-objective algorithm in SMS-EMOA [21], and so on. At the same time, proper CHT can help these efficient MOEAs to balance objective optimization and constraint satisfaction when solving CMOPs. Penalty-based methods [22] are the most widely used constraint handling methods for CMOPs where each infeasible solution is penalized based on the combination between specific penalty factors and their constraint violation values. However, the penalty factors are usually problem-independent parameters, and thus the generalization abilities of these methods are relatively weak [23]. By contrast, constraint domination principle (CDP) [24] where no user-defined parameter needs to be explicitly determined attracts great attention for solving CMOPs. Moreover, for balancing the search focus between objectives and constraints during the whole optimization, multi-objective optimization methods [25] have also been employed to handle constraints.

As we all know, no matter what kind of CHT is adopted in the constrained MOEAs (CMOEAs) to deal with CMOPs, a large number of function evaluations (FEs) are needed to obtain an acceptable PF. When the values of the objective and constraint functions at a given input are only available after running expensive simulations (for example, one simulation of a typical computational electromagnetics may take 20 min [23]), the computational cost of applying CMOEAs to solve such kind of CMOPs (also named as expensive constrained multi-objective optimization problems, ECMOPs) is unaffordable. Therefore, surrogates (also known as meta-models), which are computationally cheap models, are used to assist the CMOEAs to obtain satisfactory PF under an acceptable computational budget when solving ECMOPs. Generally, there are two classical combinations between surrogates and CMOEAs, i.e., surrogate-assisted prescreening (SAP) and surrogate-based local search (SBLS). For SAP, the surrogates, which are constructed for representing certain expensive functions, are employed to select a promising solution from many candidate solutions generated by evolutionary operations. This means that many unnecessary expensive evaluations are replaced by surrogates and the consumed number of expensive evaluations will be decreased dramatically. For SBLS, specific local optimization problem is constructed based on the characteristics of specific local samples and surrogate predictions, and thus the optimizer is employed to search the optimal solution of the local optimization problem in the corresponding local regions for accelerating the local exploitation. Over recent years, the most commonly used surrogates including radial basis function (RBF) [26] and Gaussian Processes (GP, also referred to as Kriging) [27]. Currently, these surrogate-assisted CMOEAs can also be classified into three types, i.e., indicator-based [28], dominance-based [29] and decomposition-based methods [30]. Related literature review can be found in Section 2. The indicator-based methods involve multiple complex integrals which result in high computational cost. The dominance-based methods emphasize balancing the convergence and feasibility during the optimization, while diversity is usually not explicitly considered and well maintained. Moreover, although the balance among convergence, feasibility and diversity can be effectively kept in the existing decomposition-based methods, they adopt the same search strategy for all subproblems during the whole optimization process. This means that these algorithms cannot arrange specific search strategy based on the real-time optimization state of each subproblem, which may lead to inappropriate guidance for search directions of subproblems during the optimization process. For instance, there are ten sub-problems in the current optimization, in which the first four and middle five subproblems are associated with feasible and infeasible solutions respectively, and the last sub-problem has no associated solution. At this point, the designed algorithm in this paper will conduct convergence-driven search, feasibility-driven and diversity-driven strategies for the first four, the middle five and the last one subproblems, respectively. Then the appropriate search direction can be assigned to each subproblem adaptively. However, the existing algorithms will perform the same search strategy for all the subproblems, which inevitably wastes part of the search resources on some subproblems since their suitable search directions are different. Moreover, the landscape of PF of different problems are inconsistent, and the weak real-time adaptability of the existing algorithms for different sub-problems will reduce their robustness for different problems to a certain extent.

In this paper, we propose an adaptive surrogate-assisted MOEA/D framework based on optimization states of subproblems, denoted as ASA-MOEA/D, for solving ECMOPs. Specifically, three different search strategies based on current optimization status are adaptively implemented to adjust the search of subproblems, and the both two combinations between RBF and MOEA/D, i.e., RBF-assisted prescreening and RBF-based local search, are effectively integrated in ASA-MOEA/D. The main contributions of this paper can be summarized as follows.

• An adaptive search framework for different optimization states of subproblems is constructed, in which three search strategies, i.e., feasibility-driven local search (FDLS), convergence-driven local search (CDLS) and diversity-driven global search (DDGS) are respectively employed to achieve targeted search based on three corresponding optimization states of subproblems.

• In FDLS, the orthogonal distance difference and constraint satisfaction information are effectively integrated to locate the feasible region. In CDLS, the RBF prediction values of the aggregated objectives and constraints are utilized to obtain better feasible solution. In DDGS, the RBF prediction is employed to prescreen the candidate solutions when no solution is associated with the current subproblem.

• To alleviate the misleading brought by the approximation errors of surrogates, an accuracy improving search strategy (AISS) is designed to identify the specific solution with the largest prediction uncertainty in these neighboring subregions when there exists at least one solution in current population that is constraint-dominated by the current offspring solution.

• Systematic experiments on three classical test suites show that ASA-MOEA/D with tchebycheff approach achieves the best performance in terms of two typical performance indicators (HV [31] and IGD [32]) among four state-of-the-art algorithms, i.e., C-MOEA/D [33], C-TAEA [34], SBMO(MSP + EI) [35], SMES-RBF [29], and the ASA-MOEA/D with penalty-based boundary intersection approach.

The remainder of this paper is organized as follows. Section 2 briefly introduces the related work and background techniques including multi-objective decomposition methods (MOP Decomposition), RBF and CHT. In Section 3, the proposed method is presented in detail. The experimental studies are presented and discussed in Section 4. Finally, Section 5 concludes this paper.