Computational Intelligence and Neuroscience

Volume 2019, Article ID 5126239, 27 pages

https://doi.org/10.1155/2019/5126239

## An Opposition-Based Evolutionary Algorithm for Many-Objective Optimization with Adaptive Clustering Mechanism

Zhejiang University of Technology, Hangzhou, Zhejiang 310023, China

Correspondence should be addressed to Wan Liang Wang; nc.ude.tujz@lwwtujz

Received 7 January 2019; Revised 14 February 2019; Accepted 9 April 2019; Published 2 May 2019

Academic Editor: Michele Migliore

Copyright © 2019 Wan Liang Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Balancing convergence and diversity has become a key point especially in many-objective optimization where the large numbers of objectives pose many challenges to the evolutionary algorithms. In this paper, an opposition-based evolutionary algorithm with the adaptive clustering mechanism is proposed for solving the complex optimization problem. In particular, opposition-based learning is integrated in the proposed algorithm to initialize the solution, and the nondominated sorting scheme with a new adaptive clustering mechanism is adopted in the environmental selection phase to ensure both convergence and diversity. The proposed method is compared with other nine evolutionary algorithms on a number of test problems with up to fifteen objectives, which verify the best performance of the proposed algorithm. Also, the algorithm is applied to a variety of multiobjective engineering optimization problems. The experimental results have shown the competitiveness and effectiveness of our proposed algorithm in solving challenging real-world problems.

#### 1. Introduction

Over the last two decades, evolutionary algorithm (EA) has been proven to be prevalent and efficient to solve real-world optimization problem [1]. Some of these well-known methodologies include genetic algorithms (GAs) [2], evolution strategies (ES) [3], and ant colony optimization (ACO) [4]. However, real-world optimization problems always involve multiple objectives, which means there is no single solution when considering multiple objectives as the goal of the optimization process [5]. In this case, the solutions for a multiobjective problem(MOP), which is the main focus of the algorithm, represent the trade-offs between the objectives due to the nature of such problems [6].

For the evolutionary approach to address multiobjective optimization, which is called multiobjective evolutionary algorithm (MOEA), different varieties of the algorithms have been proposed in recent years. Among them, the algorithms inspired by nature have drawn the attention like improving strength-Pareto evolutionary algorithm (SPEA2) [7], nondominated sorting genetic algorithm version 2 (NSGA-II) [8], multiobjective particle swarm optimization (MOPSO) [9], multiobjective moth-flame algorithm [10], and multiobjective ant lion optimizer [11].

There is no doubt that MOEA has been proven to be prevalent and efficient to solve optimization problem with less than three objectives [1]. However, with the gradually rising scale of data and the pluralism of target requirements, many real-world optimization problems containing more than three objectives named “many-objective problems” (MaOPs) [12, 13] are widely appearing in engineering [14], traffic [15], and water [16]. Unfortunately, the effectiveness of previous MOEAs tends to deteriorate dramatically with the increase in the number of objectives, which has been verified in [17, 18]. This can be attributed to the situation that almost all solutions in a population become nondominated with one another, and the conflict between convergence and diversity becomes aggravated with the increasing number of objectives in MaOPs [19, 20]. Moreover, computational complexity of calculating some performance metrics and the representation and visualization of the trade-off surface are also the difficulties of MaOPs. To overcome these drawbacks, a series of many-objective evolutionary algorithms (MaOEAs) have been proposed to address these optimization problems with more than three objectives. In summary, the proposed algorithms can be roughly classified into the following four types.

##### 1.1. New Domination Relation-Based Approach

As the selection criterion based on the standard dominance relationship fails to distinguish solutions in MaOPs, a number of new domination principles have been proposed to adaptively discretize the Pareto-optimal front, for instance, the *ϵ*-dominance [21, 22], CDAS-dominance [23, 24], *α*-dominance [25], fuzzy Pareto dominance [26], *L*-dominance [27], cone-domination [28, 29], and grid-based evolutionary algorithm (GrEA) [30]. Furthermore, the recently proposed generalization of Pareto optimality (GPO) [31] expands the dominance area of solutions to enhance the scalability of existing Pareto-based algorithms [32] and employs a shift-based density estimation strategy (SDE) into the dominance-based criterion, and *θ*-dominance [33, 34]introduces a new dominance relation to rank solutions, which all have been proven to be more effective than the original Pareto dominance relation. In a word, these algorithms are proposed with several variants of the Pareto dominance to enhance the selection pressure toward the PF. However, the drive toward a more aggressive selection pressure could make diversity maintenance more difficult in these new dominance-based MaOEAs.

##### 1.2. Indicator-Based Approach

Aiming to obtain a desired ordering among the representative PF approximations, indicator-based MOEAs have been widely studied, for example, the indicator-based evolutionary algorithm (IBEA) [35], SMS-EMOA [36], the fast hypervolume-based evolutionary algorithm (HypE) [37], DNMOEA/HI [38], R2 indicator based [39], [40], stochastic ranking algorithm based on multiple indicators (SRA) [41], and IGD indicator-based evolutionary algorithm (MaOEA/IGD) [42]. However, the computational cost of some performance calculation could be prohibitively expensive as the number of objectives increases.

##### 1.3. Decomposition-Based Approach

These algorithms always decompose a MaOP into several single-objective optimization problems and use aggregation functions to differentiate many-objective solutions. Among the methods that use a set of weight vectors to generate multiple aggregation functions, multiobjective evolutionary algorithm based on decomposition (MOEA/D) [43] is the most representative algorithm, which aggregates the objectives of an MOP into an aggregation function with a unique weight vector. Around the MOEA/D, several variants have been proposed to strike a better balance between convergence and diversity such as I-DBEA [44], MOEA/D-DD [45], MOEA/D-DU [46], and MOEA/D-LWS [47]. However, the situation that the number of scalarizing functions is usually very limited might cause difficulties for diversity maintenance of the solution especially when compared with the exponentially increasing objective space.

##### 1.4. Reference Set-Based Approach

The algorithms of this category use a set of reference solutions to measure the quality of solutions. Thus, the search process is guided by the solutions in the reference solution set. Praditwong and Yao [48] and Wang [49] proposed a novel two archive algorithm (TAA) and its improved version (Two_Arch2), in which the convergence archive (CA) can be seen as an online-updated real reference set. The recently proposed vector angle-based evolutionary algorithm (VaEA) [50] uses the population as the reference set to dynamically guide the evolutionary process. As for the algorithm using virtual reference set, NSGA-III [51, 52]is predominant which employs a set of predefined reference points to manage the diversity of the candidate solutions. Hereafter, a number of algorithms have been proposed with the reference point or vector such as the reference vector-guided evolutionary algorithm for many-objective optimization (RVEA) [53], reference vector-guided evolutionary Pareto evolutionary algorithm with reference direction (SPEAR) [54], and many-objective evolutionary optimization based on reference points (RPEA) [55].

The algorithm from the third category and fourth category becomes particularly prevalent for many-objective optimization, which could be attributed to the low cost to achieve a representative subset of the entire PF especially in a limited population size. However, most algorithms belong to these categories simply use the angle or distance solely to measure the quality of the population members with the reference set, which may lose some good solutions due to their simplex selection mechanism. Furthermore, it has been logically proved by the No Free-Lunch (NFL) theorem [56] that none of these algorithms is able to solve all optimization problems, which allows the researchers to propose new methods or improve the current algorithms for better solving the problems [33, 57]. Therefore, this paper proposes an opposition-based multiobjective evolutionary algorithm with an adaptive clustering mechanism, in short named OBEA to strengthen the selection mechanism through comprehensive consideration of the angle and the distances. The main properties of OBEA can be summarized as follows:(i)A new initialization approach is designed with the assistance of the opposition-based learning (OBL) to generate the population. Due to the fact that random initialization lowers the chance of sampling better regions in algorithms, here we use the OBL to initialize the populations in stand of the previous random method. Moreover, opposition-based learning is also adapted in the evolutionary process with the aim of enhancing the probability of obtaining the better solutions.(ii)An adaptive clustering strategy is integrated in this algorithm. In this proposed strategy, the acute angle and perpendicular distance between the candidate solutions and the reference vectors are combined through an adaptive mechanism to cluster the candidate. Moreover, a novel selective approach is designed to dynamically select the individual with comprehensive consideration of the balance convergence and diversity.

Furthermore, an extensive comparison between the proposed OBEA with nine algorithms is implemented on 60 instances of 14 test problems taken from two well-known test suites. The results indicate that OBEA is a very promising alternative for many-objective optimization. The rest of this paper is organized as follows. Section 2 introduces the background knowledge, and details of the proposed OBEA are described in Section 3. Section 4 presents the numerical results of OBEA on benchmark and the detailed analysis of the proposed algorithm together with nine MaOEAs. Finally, conclusions and future work are given in Section 5.

#### 2. Background

In this section, the main components of multiobjective optimization problem (MOP) are given first, which involve the basic knowledge of optimization and Pareto dominate. Next, a brief description of the reference vector is given, which is used as the underlying mechanism for solving many-objective optimization problems.

##### 2.1. Multiobjective Problem

Generally, a multiobjective optimization problem can be stated as follows:where is the decision vector that satisfies and stands for the decision space. The objective function vector consists of *m* () objectives and refers to the objective space.

*Definition 1. *Given two vector , **x** is *Pareto dominate ***y** (denoted as ) if and only if for each , and , .

*Definition 2. *A decision vector is said to be *Pareto optimal*, if and only if there is no .

*Definition 3. *The set of Pareto-optimal solutions (PS) is defined as .

*Definition 4. *The Pareto-optimal front (PF) is defined as .

##### 2.2. Reference Vector

As an underlying mechanism throughout the algorithm, OBEA uses Das and Dennis’s systematic approach [58], and adaptation of this approach generates the reference points and thus forms the reference vectors. The original Das and Dennis’s method places points on a normalized hyperplane, where the number of reference points depends on the dimension of objective space *m* and positive integer *H*. The equation can be described as follows:

However, the number of reference points would rapidly increase when *m* is a relatively large number. To address the drawback of the computational burden of the reference point, a number of new approaches have been proposed [45, 59]. OBEA utilizes the two-layered reference point generation approach, as suggested in [33]. The hyperplane is divided into two parts, which are the boundary and inner layers, as shown in Figure 1. The detail of implementation is as follows: