Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 1435463, 18 pages

https://doi.org/10.1155/2018/1435463

## An Indicator and Decomposition Based Steady-State Evolutionary Algorithm for Many-Objective Optimization

^{1}Department of Information Science and Engineering, Northeastern University, Shenyang 110819, China^{2}School of Information Science and Engineering, Central South University (CSU), Changsha 410083, China^{3}School of Mechanical Engineering, Shenyang Jianzhu University, Shenyang 110168, China

Correspondence should be addressed to Fei Li

Received 6 August 2017; Revised 11 December 2017; Accepted 22 January 2018; Published 11 March 2018

Academic Editor: Giuseppe Vairo

Copyright © 2018 Fei Li 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

An indicator based selection method is a major ingredient in the formulation of indicator based evolutionary multiobjective optimization algorithms. The existing classical indicator based selection methodologies have demonstrated an excellent performance to solve low-dimensional optimization problems. However, the indicator based evolutionary multiobjective optimization algorithms encounter enormous challenges in high-dimensional objective space. Our main purpose is to explore how to extend the indicator to handle many-objective optimization problems. After analyzing the indicator, the objective space partition strategy, and the decomposition method, we propose a steady-state evolutionary algorithm based on the indicator and the decomposition method, named, -MOEA/D, to obtain well-converged and well-distributed Pareto front. The main contribution of this paper contains two aspects. (1) The convergence and diversity for the indicator based selection are analyzed. Two improper selection situations will be properly solved via applying the decomposition method. (2) According to the position of a new individual in the steady-state evolutionary algorithm, two different objective space partition strategies and the corresponding selection methods are proposed. Extensive experiments are conducted on a variety of benchmark test problems, and the experimental results demonstrate that the proposed algorithm has competitive performance in comparison with several tailored algorithms for many-objective optimization.

#### 1. Introduction

A large number of evolutionary multiobjective optimization algorithms (EMOAs) have been introduced to solve multiobjective optimization problems (MOPs) [1–5]. Most of these algorithms have demonstrated their excellent performance to deal with MOPs involving two or three objectives. However, most of optimization problems, such as water resource management problem [6], general aviation aircraft design problem [7], and hybrid electric vehicle controller design problem [8], often relate to many-objective optimization problems [9] (MaOPs). The traditional EMOAs face substantial difficulties when addressing MaOPs. There is enormous amount of researches discussing most existing methodologies to solve them. It has been one of the major research aspects as the detailed survey of MaOPs [10].

Convergence is to find a set of solutions as close as possible to the Pareto front while diversity is to obtain well-distributed solutions. Balancing convergence and diversity is a key issue for solving MaOPs. In order to obtain the better performance, several EMOAs have been presented in the literature which can be divided into four categories.

The idea of the first category is based on the Pareto dominance relation. In the dominance based methods, the Pareto dominance can be defined as the preference of decision maker. The popular EMOAs, such as nondominated sorting genetic algorithm II (NSGA-II) [11] and multiple particle swarm optimizer (MOPSO) [12], are based on the Pareto dominance. These algorithms can deal with MOPs. However, the selection pressure of Pareto dominance is severe loss with the dimension of the objective space increasing. Therefore, there is a large amount of studies to modify the dominance relations or adopt the secondary metric. Fuzzy dominance [13, 14] and preference inspired method [15] have been deployed for handling MaOPs. However, the final obtained Pareto solutions may present an excellent convergence in objective space but with the poor diversity.

The second category is the decomposition based selection method. The method is to decompose an MOP into many single-objective optimization subproblems through a scalarizing function. It aims to optimize these subproblems in a collaborative manner by evolving the population. C-MOGA [16] and MOEA/D [17] are the most representative of this sort. During the past few years, as a major framework to design EMOAs, the decomposition based selection method has spawned a large number of research works, for example, incorporating self-adaptation mechanisms in reproduction [18] and hybridizing with the swarm intelligence [19, 20].

The third avenue is known as the indicator based approaches, which merge the convergence and diversity into an indicator. The indicator based evolutionary algorithms employ a single performance indicator to provide a desired ordering among individuals that represent Pareto front approximations. The hypervolume [21] is probably the most popular indicator ever adopted due to its satisfactory theoretical properties. Some hypervolume based EMOAs have been established, such as indicator-based evolutionary algorithm (IBEA) [22] and multiobjective covariance matrix adaptation evolution strategy (MO-CMA-ES) [23]. However, the computational cost of hypervolume hinders these algorithms for MaOPs [24].

Recently, several new performance indicators called [25, 26], [27], and [28–31] have been proposed. More specifically, is composed of slight modifications of and performance indicators via introducing the averaged Hausdorff distance. Its computation cost is lower than that of the hypervolume, and it can handle outliers as well, which makes it attractive for assessing performance of EMOAs [32]. Meanwhile, it has been shown in [29, 33] that prefers evenly spread solutions along the Pareto front for biobjective and three-objective optimization problems, respectively. However, the main limitation of the indicator is how to produce the well-converged and well-distributed candidate solutions in high-dimensional objective spaces. Nowadays, a new performance indicator called the indicator is proposed in [34] to compare approximation sets based on a set of utility functions [35]. The indicator is weakly monotonic and performs a lower computational overhead than the hypervolume. Due to these characteristics, the indicator is recommended for dealing with MaOPs [32]. MOMBI [32] and MOMBI-II [36] have been proposed in the context of MaOPs.

It is worth noting that, unlike the three above-mentioned categories, the objective space partition strategy can be regarded as the fourth category to handle MaOPs. The representative EMOAs are MOEA/D-M2M [37] and MOEA/DD [38]. The motivation of this category is to balance convergence and diversity in each subspace. In comparison to the traditional decomposition based methods [17], these EMOAs can achieve a better performance via introducing the sophisticated selection methods.

In other words, the generational and steady-state schemes are two commonly used reproduction operators to produce the offspring population. Most of generational EMOAs have been deeply studied other than the steady-state EMOAs, especially in -EMOAs. There is still huge room for improvement especially for the steady-state evolutionary algorithm to resolve MaOPs. Meanwhile, MOMBI-II [36] and MOEA/DD [38] are suitable for different problems. This phenomenon motivates us on the merits of both selection strategies for further research to balance convergence and diversity. This paper mainly focuses on the indicator based steady-state evolutionary algorithm which adopts the modified Tchebycheff and penalty-based boundary intersection (PBI) decomposition methods to balance convergence and diversity for MaOPs. The main contributions of this paper are as follows.

(1) It is the first time to combine the indicator and the decomposition method into a steady-state evolutionary algorithm to address MaOPs. To be specific, we find that the indicator based selection method is not always proper for the steady-state evolutionary algorithm. It is harmful for the population diversity when purely applying the indicator to handle MaOPs. Therefore, we employ the objective space partition strategy and the PBI decomposition method to achieve a balance between convergence and diversity.

(2) The selection procedure of the steady-state evolutionary algorithm is to delete a candidate solution from the combined candidate solutions which contain the parent population and an offspring individual. Two types of the selection approaches will be introduced to delete a candidate solution in terms of the ideal point. On the one hand, we first repartition the objective space into a number of subspaces if the position of the ideal point changes. Then, we adopt the indicator and the decomposition method to prune the combined candidate solutions. On the other hand, we cluster the offspring candidate solution to the nearest reference vector if the ideal point remains unchanged. Then, the decomposition method is adopted to discard a solution which has the worst performance in the most crowded subspaces.

The rest of this paper is organized as follows. Section 2 provides the background and motivation of this paper. Section 3 is devoted to the description of our proposed -MOEA/D for MaOPs. Comprehensively experimental design and experimental results are provided in Sections 4 and 5. Finally, Section 6 concludes this paper.

#### 2. Background and Motivation

In this section, some basic definitions and related works about -MOEA/D are firstly introduced. Then, the motivation of -MOEA/D will be illustrated in detail.

##### 2.1. Basic Definitions

Without loss of generality, a minimization multiobjective optimization problem can be formulated as follows:where is the vector of decision variables, is the dimension of solution space, is the th objective, is the number of objectives, and is the vector of dimensional objectives. Given two vectors and in the objective space, we say that dominates if , for all , and . Pareto set (PS) corresponding to all of nondominated solutions in decision space and Pareto front (PF) is in objective space.

In [35], the indicator was first proposed to assess the convergence and diversity of a set of candidate solutions . According to [34], the definition of the indicator can be given as follows.

*Definition 1. *For a set of reference vectors and the modified Tchebycheff utility function, the indicator of a solution set can be defined as follows:

*Definition 2. *The contribution of a solution to the indicator can be of interest to assess the performance of each individual, which is defined as follows:where is called a set of reference vectors, is an ideal point, and is the individual of population .

In this paper, we adopt the penalty-based boundary intersection (PBI) method, due to its promising performance for many-objective optimization reported in MOEA/DD [38]. Without loss of generality, the PBI decomposition method can be given as follows:where is the convergence distance metric, denotes the diversity distance metric, and is an important parameter to balance convergence and diversity.

##### 2.2. Related Works

In this subsection, we would illustrate the representative indicator and decomposition based selection methods which are based on a set of reference vectors.

(i) MOMBI [32] and MOMBI-II [36] are two representative -EMOAs which abandon the Pareto dominance concept for MaOPs. The quality of a solution is fully determined by a set of reference vectors. The achievement scalarizing function is first introduced into the -EMOAs. MOMBI-II has adopted statistical information of previous generations to normalize the candidate solutions. However, the fast ranking strategy scarcely considers the relationship between the adjacent ranks. The convergence is well while the diversity will be destroyed.

(ii) DBEA [7] is an excellent and practical steady-state EMOA for MaOPs which not only considers the benchmark test problems but also takes into account three constrained engineering design optimization problems. The innovation of DBEA contains two parts: one is a simple selection procedure without introducing any penalty parameter, where the diversity distance metric has a precedence over the convergence distance metric, and the other is the normalization method, which is based on the corner sort method.

(iii) MOEA/D-M2M [37] is the first paper which introduces a set of weight vectors to divide the objective space into a large number of subspaces. Different weight vectors are used to specify different subpopulations for approximating a small segment of the whole Pareto front. The algorithm shows a better performance to deal with biobjective or three-objective optimization problems.

(iv) RVEA [39] can be seen as a generational EA to produce offspring and the steady-state selection strategy based on a set of reference vectors. The adaptive angle-penalized distance (APD) metric and a reference vector updated strategy are firstly introduced to balance convergence and diversity and to deal with scaled MaOPs. Due to the excellent adaptive reference vector updated strategy, we have embedded this strategy into the proposed -MOEA/D algorithm.

(v) MOEA/DD [38] is on the merit of Pareto dominance and decomposition method to balance convergence and diversity. The main contribution contains three parts. Firstly, the mating restriction strategy considers neighboring subspace and the whole population. Secondly, the efficient nondominated level update approach is proposed for steady-state EMOAs. Finally, the density of the population is calculated by the local niche count of a subspace. MOEA/DD [38] is currently an excellent steady-sate EA for MaOPs.

##### 2.3. Motivation

Achieving a balance between convergence and diversity is the cornerstone of the indicator and decomposition based evolutionary algorithms. The motivation of this paper is on the merit of the indicator and decomposition method to balance convergence and diversity for the steady-state evolutionary algorithm to address MaOPs. Two main aspects should be taken into account.

(1) As suggested in [36], the indicator naturally balances convergence and diversity during the evolutionary procedure. However, the selection pressure for solving MaOPs hardly quantitatively balances convergence and diversity simply depending on the indicator. Meanwhile, the indicator gives an excessive priority to the convergence requirement while the objective space partition and decomposition methods pay more attention to diversity other than convergence. Therefore, how to reasonably balance convergence and diversity on the benefit of the indicator and the decomposition method is vital for solving MaOPs.

(2) As for the steady-state EAs, only an individual is created and a candidate solution will be deleted from the combined candidate solutions in each generation. The key issue of the steady-state EA is to determine which candidate solution will be substituted by an offspring. Considering the position information between the offspring and the ideal point, the objective space partition strategy should be illustrated in detail.

First and foremost, the indicator [34] which can be seen as the mutual preference between a set of reference vectors and candidate solutions is a vital ingredient in EMOAs for MaOPs. Most of researchers [32, 36, 40–42] believe that the indicator obtains a desirable performance which combines convergence and diversity. Let us consider an example as shown in Figure 1(a), where a candidate solution needs to be deleted from the combined candidate solutions. The indicator is adopted to prune the combined population and will be deleted from the population. It is a proper situation for the indicator based selection method. However, we find that purely adopting the indicator is not always reasonable as shown in Figure 1(b). In this case, although the indicator based selection mechanism intends to achieve a balance between convergence and diversity, the selected candidate solutions fail to keep the population diversity especially in high-dimensional objective space. The main reason is that the indicator based selection method will discard which is located in the isolated subspace. To relieve this effect, whose position is located in the most crowded subspace will be discarded while will survive. Based on the above description and discussion, it is interesting to note that combining the indicator with decomposition method is a potential improvement strategy to balance convergence and diversity for MaOPs.