The Scientific World Journal

Volume 2015 (2015), Article ID 439307, 10 pages

http://dx.doi.org/10.1155/2015/439307

## A Novel Multiobjective Evolutionary Algorithm Based on Regression Analysis

^{1}School of Computer, China University of Geosciences, Wuhan 430074, China^{2}Department of Mechanical & Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UK

Received 23 June 2014; Revised 15 September 2014; Accepted 30 December 2014

Academic Editor: Shifei Ding

Copyright © 2015 Zhiming Song 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

As is known, the Pareto set of a continuous multiobjective optimization problem with objective functions is a piecewise continuous ()-dimensional manifold in the decision space under some mild conditions. However, how to utilize the regularity to design multiobjective optimization algorithms has become the research focus. In this paper, based on this regularity, a model-based multiobjective evolutionary algorithm with regression analysis (MMEA-RA) is put forward to solve continuous multiobjective optimization problems with variable linkages. In the algorithm, the optimization problem is modelled as a promising area in the decision space by a probability distribution, and the centroid of the probability distribution is ()-dimensional piecewise continuous manifold. The least squares method is used to construct such a model. A selection strategy based on the nondominated sorting is used to choose the individuals to the next generation. The new algorithm is tested and compared with NSGA-II and RM-MEDA. The result shows that MMEA-RA outperforms RM-MEDA and NSGA-II on the test instances with variable linkages. At the same time, MMEA-RA has higher efficiency than the other two algorithms. A few shortcomings of MMEA-RA have also been identified and discussed in this paper.

#### 1. Introduction

Evolutionary algorithm has become an increasingly popular design and optimization tool in the last few years [1]. Although there have been a lot of researches about evolutionary algorithm, there are still many new areas that needed to be explored with sufficient depth. One of them is how to use the evolutionary algorithm to solve multiobjective optimization problems. The first implementation of a multiobjective evolutionary algorithm dates back to the mid-1980s [2]. Since then, many researchers have done a considerable amount of works in the area, which is known as multiobjective evolutionary algorithm (MOEA).

Because of the ability to deal with a set of possible solutions simultaneously, evolutionary algorithm seems particularly suitable to solve multiobjective optimization problems. The ability makes it possible to search several members of the Pareto-optimal set in a single run of the algorithm [3]. Obviously, evolutionary algorithm is more effective than the traditional mathematical programming methods in solving multiobjective optimization problem because the traditional methods need to perform a series of separate runs [4].

The current MOEA research mainly focuses on some highly related issues [5]. The first issue is the fitness assignment and diversity maintenance. Some techniques such as fitness sharing and crowding have been frequently used to maintain the diversity of the search. The second issue is the external population. The external population is used to record nondominated solutions found during the search. There have been some efforts on how to maintain and utilize such an external population. The last issue is the combination of MOEA and local search. Researches have shown that the combination of evolutionary algorithm and local heuristics search outperforms traditional evolutionary algorithms in a wide variety of scalar objective optimization problems [4, 6].

However, there are little researches focusing on the way to generate new solutions in MOEA. Currently, most MOEAs directly adopt traditional genetic operators such as crossover and mutation. These methods have not fully utilized the characteristics of MOP when generating new solutions. Some researches show that MOEA fails to solve MOPs with variable linkages, and the recombination operators are crucial to the performance of MOEA [7]. It has been noted that under mild smoothness conditions, the Pareto set (PS) of a continuous MOP is a piecewise continuous ()-dimensional manifold, where is the number of the objectives. However, as analyzed in [8], this regularity has not been exploited explicitly by most current MOEA.

In 2005, Zhou et al. proposed to extract regularity patterns of the Pareto set by using local principal component analysis (PCA) [9]. They had also studied two naive hybrid MOEAs. In the two MOEAs, some trial solutions were generated by traditional genetic operators and others by sampling from probability models based on regularity patterns in 2006 [10].

In 2007, Zhang et al. conducted a further and thorough investigation along their previous works in [9, 10]. They proposed a regularity model-based multiobjective estimation of distribution algorithm and named it as RM-MEDA [5]. At each generation, the proposed algorithm models a promising area in the decision space by a probability distribution whose centroid is a ()-dimensional piecewise continuous manifold. The local principal component analysis algorithm is used to build such a model. Systematic experiments have shown that RM-MEDA outperforms some other algorithms on a set of test instances with variable linkages.

In 2008, Zhou et al. proposed a probabilistic model based multiobjective evolutionary algorithm to approximate PS and PF (Pareto front) for a MOP in this class simultaneously and named the algorithm as MMEA [11]. They proposed two typical classes of continuous MOPs as follows. One class is that PS and PF are of the same dimensionality while the other one is that PF is a ()-dimensional continuous manifold and PS is a continuous manifold with a higher dimensionality. There is a class of MOPs, in which the dimensionalities of PS and PF are different so that a good approximation to PF might not approximate PS very well. MMEA could promote the population diversity both in the decision spaces and in the objective spaces.

Modeling method is a crucial part for MOEA because it determines the performance of the algorithms. Zhang et al. built such a model by local principal component analysis (PCA) algorithm [5]. The test results show that the method has great performance over some instances with linkage variables. However, there are still some shortcomings about the method. The first shortcoming is that RM-MEDA needs extra CPU time for running local PCA at each generation. The second one is that the model is just linear fitting for all types of PS, including the one with nonlinear linkage variables, which enable that the result may be not accurate.

In the paper, we proposed a model-based multiobjective evolutionary algorithm with regression analysis, which is named as MMEA-RA. In MMEA-RA, a new modeling method based on regression analysis is put forward. In the method, least squares method (LSM) is used to fit a 1-dimensional manifold in high-dimensional space. Because least squares can fit any type of curves through its model, the shortcomings of RM-MEDA can be avoided, especially for the instances with nonlinkage variables.

The rest of this paper is organized as follows. After defining the continuous multiobjective optimization problem in Section 2, the new model of multiobjective evolutionary algorithm based on regression analysis is put forward in Section 3. Then, a description of the test cases for MMEA-RA follows in Section 4. After presenting the results of the tests, the performance of MMEA-RA is analyzed and some conclusions are given in Section 5.

#### 2. Problem Definition

In this paper, the continuous multiobjective optimization problem is defined as follows [5]:where is the decision space and is the decision vector. consists of real-valued continuous objective functions . is the objective space.

Let and be two vectors, and is said to dominate , denoted by , if for all , and . A point is called (globally) Pareto optimal if there is no such that . The set of all Pareto-optimal points, denoted by PS, is called the Pareto set. The set of all Pareto objective vectors is called the Pareto front, denoted by PF.

#### 3. Algorithm

##### 3.1. Basic Idea

Under certain smoothness assumptions, it can be induced from the Karush-Kuhn-Tucker condition that the PS of a continuous MOP defines a piecewise continuous ()-dimensional manifold in the decision space [12]. Therefore, the PS of a continuous biobjective optimization problem is a piecewise continuous curve in .

The population in the decision space in a MOEA for (1) will hopefully approximate the PS and is uniformly scattered around the PS as the search goes on. Therefore, we can envisage the points in the population as independent observations of a random vector whose centroid is the PS of (1). Since the PS is a ()-dimensional piecewise continuous manifold, can be naturally described by where is uniformly distributed over a piecewise continuous ()-dimensional manifold, and is an -dimensional zero-mean noise vector. Figure 1 illustrates the basic idea.