ACDB-EA: Adaptive convergence-diversity balanced evolutionary algorithm for many-objective optimization

Abstract

Recently, evolutionary algorithms (EAs) have shown their strong competitiveness in handling many-objective optimization problems (MaOPs) with different Pareto fronts (PFs). However, maintaining convergence and diversity simultaneously in high-dimensional problems can be further explored. This paper suggests an adaptive convergence-diversity balanced evolutionary algorithm (ACDB-EA) to handle the above issue, which maintains balance between convergence and diversity adaptively during the evolutionary process. In the proposed algorithm, a novel diversity maintenance mechanism based on the global and local diversities is developed to promote the diversity by considering them collaboratively. To be specific, the average similarity and the maximal similarity represent respectively the global and local diversities of the solution, where the similarity between two solutions is defined as the cosine similarity between their objective vectors. In the environmental selection, the proposed adaptive convergence-diversity balanced strategy is used to adjust weights of convergence (defined as the $L_2$ norm in the objective space), global diversity and local diversity according to the population adaptively. Under this strategy, each solution produces a score and the solution with the highest score will enter the next generation, which means it acquires the optimal performance in terms of convergence and diversity. In each iteration, scores of candidate solutions will be recalculated to continuously search for the most suitable one, which strengthens the selection pressure toward the true PFs. We conduct experimental study on 111 benchmark testing instances with 2-20 objectives. The proposed method is shown to be superior to seven state-of-the-art algorithms in maintaining balance between convergence and diversity.

Introduction

Many real-world applications have been described as many-objective optimization problems (MaOPs), such as engineering design [1], storage problem of blockchain [2], water allocation problem [3]. In this paper, we consider the following MaOP:

$$\text{min}f\left(\mathit{x}\right)={(f_1\left(\mathit{x}\right),f_2\left(\mathit{x}\right),\dots ,f_m\left(\mathit{x}\right))}^T\subseteq {\mathbb{R}}^m$$$$\text{s.t.}\mathit{x}\in \mathit{\Omega }\subseteq {\mathbb{R}}^n$$

where $\mathit{\Omega }\subseteq {\mathbb{R}}^n$ is called the decision space and $\mathit{x}={(x_1,x_2,\dots ,x_n)}^T$ indicates a candidate solution. ${\mathbb{R}}^m$ is the objective space and $f:\mathit{\Omega }\to {\mathbb{R}}^m$ consists of $m$ ($m\ge 4$) objective functions to be minimized.

Many-objective evolutionary algorithms (MaOEAs) are population-based approaches [4], [5], [6], which aim to acquire a set of nondominated solutions achieving well convergence and diversity¹ on the true Pareto front (PF) of MaOPs. During the last few decades, plenty of MaOEAs have been proposed and verified to be suitable for MaOPs [7], [8], [9], [10]. However, their ability of handling high-dimensional problems can be further explored, especially for Pareto-based MaOEAs. The main difficulty is that solutions become incomparable with each other in high-dimensional problems, which means the proportion of nondominated solutions increases. And such situation becomes worse with the increasing number of objectives, resulting in weaker selection pressure [11], [12]. The second is the difficulty of maintaining population diversity. Existing approaches for preserving the diversity may not be efficient in the high-dimensional objective space [13], such as $k$th nearest distance [14] and crowding distance [15]. To tackle these two issues, researchers have developed many algorithms, and most of them are classified as three categories.

The first category is known as the Pareto dominance-based MaOEAs, which enhances the selection pressure by modifying the definition of classical Pareto dominance, making solutions be able to compare with each other. For instance, fuzzy dominance [16], $ϵ$-dominance [17], $\theta$-dominance [18] adopt modified Pareto dominance to choose solutions. Tian et al. [19] designed a novel dominance relation to obtain better balance between convergence and diversity based on an adaptive technique. Qiu et al. [20] proposed fractional dominance to strengthen convergence by comparing the value of each objective between two solutions. Besides, another method is to combine the classical dominance and an additional convergence-related metric. Zhang et al. [21] proposed knee point driven evolutionary algorithm (KnEA) to enhance the convergence performance.

The second category covers the indicator-based MaOEAs, which adopts indicators as the selection criterion to assess nondominated solutions and guide the evolutionary procedure. The R2 indicator [22], the S-metric [23], $I_{\varepsilon }$ indicator [24], and the hypervolume (HV) indicator [25] are widely used indicators. For example, HV-based MaOEAs [26], [27], [28] evaluate solutions according to their contributions to the HV value. Moreover, arbitrary indicators can be integrated into the indicator-based evolutionary algorithm (IBEA) [24].

The third category represents the decomposition-based MaOEAs, which decomposes a complex MaOP into simpler multiobjective subproblems [29] or a group of single-objective optimization problems (SOPs) [30], and then solve them collaboratively. Two representative algorithms are MOEA/D [30] and NSGA-III [13]. In MOEA/D, each subproblem is optimized based on the information on its several neighboring subproblems, which enables to reduce computational complexity. In NSGA-III, a set of reference points are adopted to decompose the objective space, which promotes the diversity. Following the basic idea of them, some environmental selection strategies have been proposed. Li et al. [31] combined dominance and decomposition, while Yuan et al. [32] exploited the perpendicular distance from solutions to reference vectors in the objective space. To tackle the issue of lacking the information of true PFs beforehand, some effective approaches are proposed, such as adapting the weights during the evolutionary process [33], learning the distribution of reference vectors by the growing neural gas (GNG) network [34] and extracting reference vectors by a modified $k$-means clustering method [35].

There are also some MaOEAs that are not located in the above categories, such as determinantal point process-based algorithm [36], adaptive clustering-based algorithm [37], two-archive algorithm [38], [39] (i.e., convergence archive and diversity archive), dimension reduction and solving knowledge guided algorithm [40], voting-based algorithm [41], two-stage algorithm [42] and decision variable classification-based algorithm [43]. Ding et al. [41] proposed an effective multi-stage evolutionary strategy based on knowledge fusion. The statistical guidance vector was designed as one of the knowledge, which calculated the concentration degree of elite solutions on the basis of voting method. Li et al. [42] solved the objective optimization by a convergence stage and a diversity stage. In the convergence stage, the individuals competed with each other. In the diversity stage, the solutions with well space-filling were chosen. Liu et al. [43] developed a novel decision variable classification method to generate promising solutions, which achieved a higher probability to produce offspring with well convergence and diversity.

Although there are a lot of research on MaOEAs, handling the situation to balance convergence and diversity in high-dimensional problems still needs to be further explored [44], [45], [46]. Algorithms need to promote good diversity of the population, while guarantee high selection pressure upon the true PFs. The maximal marginal relevance (MMR) [47] is a popular approach in improving diversity, which is widely used in text retrieval [48], document summarization [47], product recommendation [49], and so on. The way of generating document summarization gives us a motivation of this paper. Further study is explored based on the above work. Specifically, when choosing solutions into the next generation, the solution which achieves well convergence towards the true PFs and contains the minimal similarity to the previously selected solutions should be chosen firstly. This way, the performance of balancing convergence and diversity can be improved to obtain satisfactory final population.

Inspired by the work in [47], we propose an adaptive convergence-diversity balanced evolutionary algorithm (ACDB-EA) to solve the problem of maintaining the balance between convergence and diversity. In the proposed algorithm, the contribution of each solution is quantitatively calculated by implementing the adaptive convergence-diversity balanced strategy, which produces a score based on convergence, global and local diversities. Solution with the highest contribution will be selected into the next generation, which means it obtains the better balance between convergence and diversity. To be specific, the convergence of a solution is defined as the $L_2$ norm in the objective space, and the diversity of a solution is measured by its similarity in the population, where the similarity between two solutions is defined as the cosine similarity between their objective vectors. In this way, the average similarity and the maximal similarity represent respectively the global and local diversities of the solution, aiming to enhance diversity. The adaptive convergence-diversity balanced strategy adjusts weights of convergence and diversity, global diversity and local diversity according to the population adaptively, which aims to make an appropriate tradeoff between convergence and diversity. To promote diversity in the objective space, a clustering strategy is also adopted in this paper. Furthermore, to enhance the quality of the mating pool, we develop a convergence-crowding-based mating strategy and an elitist solution archive. In the convergence-crowding-based mating strategy, selection of the mating parent depends on the quality of convergence and crowding. The harmonic average distance (HAD) [50] is adopted to reflect the crowding status of solutions in the objective space, which estimates the density around one solution by using the $k$-nearest neighbor distances. In elitist solution archive, elitist solutions with good convergence and diversity on each objective dimension are preserved on each generation.

The main contributions of this paper are summarized as follows.

• A convergence-crowding-based mating strategy which selects the mating parent based on convergence and crowding is proposed to balance convergence and diversity in the mating pool. To be specific, the $L_2$ norm and the HAD method are used to reflect the convergence and crowding status of solutions, respectively. This strategy gives corresponding probabilities $P_n$ and $(1-P_n)$ of choosing the mating parent based on the convergence and crowding.

• An elitist solution archive, termed ESA, is suggested to preserve elitist solutions with good convergence and diversity on each objective dimension. In the process of evolution, ESA updates at every generation to promote the mating pool, aiming to produce high-quality offsprings.

• An adaptive convergence-diversity balanced (ACDB) strategy is developed to quantitatively calculate the score of each solution based on the convergence, global diversity and local diversity. Solution with the highest score obtains the better balance between convergence and diversity. By selecting the solution with the highest score in each iteration, the proposed algorithm achieves well balance between convergence and diversity.

The remainder of this paper is shown as follows. The related work is summarized in Section 2. The preliminaries are described in Section 3. Section 4 elaborates the proposed ACDB-EA. Experimental results are presented and discussed in Section 5. In the end, the conclusions and future work are drawn in Section 6.