Cross-Searching Strategy for Multi-objective Particle Swarm Optimization Shih-Yuan Chiu, Tsung-Ying Sun, Sheng-Ta Hsieh, Student Member, IEEE, Member, IEEE and Cheng-Wei Lin
Abstract—The main difference between an original PSO (single-objective) with a multi-objective PSO (MOPSO) is the local guide (global best solution) distribution must be redefined in order to obtain a set of non–dominated solutions (Pareto front). In MOPSO, the selection of local guide for particles will direct affect the performance of finding Pareto optimum. This paper presents a local guide assignment strategy for MOPSO called cross-searching strategy (CSS) which will distribute suitable local guides for particles to lead them toward to Pareto front and also keeping diversity of solutions. Experiments were conducted on several test functions and metrics from the standard literature on evolutionary multi-objective optimization. The results demonstrate good performance of the CSS for MOPSO in solving multi-objective problems when compare with recent approaches of multi-objective optimizer. Index Terms Local guide, multi-objective particle swarm optimization (MOPSO), cross-searching strategy
I. INTRODUCTION
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ulti-objective optimization (MO) has been an active area of research in last two decade. Such problems arise in many applications, where two or more, sometimes competing and/or incommensurable objective functions have to be minimized concurrently. Since Schaffer proposed a Vector Evaluated Genetic Algorithm (VEGA) [1], [2] in 1984, many MO algorithms have then been proposed. Hajela and Lin proposed a Weight-based Genetic Algorithm (WBGA) [3], and the Multi-Objective GA (MOGA) [4] was proposed by Fonseca and Fleming. Later, Srinivas and Deb proposed a Non-dominated Sorting Genetic Algorithm (NSGA) [5] and the Niched Pareto Genetic Algorithm (NPGA) [6] was proposed by Horn et al. It introduced a binary tournament selection but does not assign a definite fitness value, but problems with more optimized objectives will influence the computational efficiency of NPGA. Several efficient strategies have been introduced based on these algorithms, such as Elitism, external repository, or archive. The Strength Pareto Evolutionary Algorithm (SPEA) [7] was proposed by Zitzler and Thiele. It introduced an elitism strategy to store an extra population that contains non-dominated solutions. New found non-dominated solutions will be compared with the extra stored population,
This work was supported under Grant (NSC95-3113-P-259-002) by the National Science Council of Taiwan, R.O.C. Authors are with the National Dong Hwa University, Hualien, Taiwan, R.O.C. (Tsung-Ying Sun as corresponding author to provide phone: +886-3-8634078; fax: +886-3-8634060; e-mail:
[email protected]).
and the better is kept. The SPEA2 [8] is an advanced version of SPEA. SPEA2 inherited the advantages from SPEA and improved fitness assignment to take both dominated and non-dominated solutions into account. In [9], Knowles et al. proposed Pareto Archive Evolution Strategy (PAES), which employs (1+1) evolution strategy (ES) and uses a mutation operator for local searches. A map of a grid is applied in the algorithm to maintain the diversity of the archive. Thus, there will be a trade-off to define the size of both the external repository and grid of the map. Deb proposed an enhanced NSGA named NSGA-II [10], [11] which employs a fast non-dominated approach to assign ranks to individuals and crowded tournament selection for density estimation. In the case of a tie in rank during the selection process, the individual with a lower density count will be chosen. As numerous MOGA approaches were proposed, many researches are interesting in PSO solving multi-objective problems [12]-[15]. The main difference between an original PSO (single-objective) with a multi-objective PSO (MOPSO) is the local guide (gbest) distribution must be redefined in order to obtain a set of non–dominated solutions (Pareto front). Hu and Eberhart proposed a dynamic neighborhood PSO [14], which optimizes only one objective at a time and uses a scheme similar to lexicographic ordering [16]. In [13], Fieldsend and Singh proposed an unconstraint elite archive named dominated tree to store the non-dominated solutions, but it’s a difficult issue for this approach to pick up a best local guide from the set of Pareto-optimal solutions for each particle of the population. A strategy for finding suitable local guides for each particle was proposed by Mostaghim and Teich named sigma method [17]. The local guide is explicitly assigned to specific particles according to the sigma value. This results in desired diversity and convergence but its still not close enough to the Pareto front. On the other hand, an enhanced archiving technique to maintain the best (non–dominated) solutions found during the course of a MO algorithm was proposed in [18]. It shows that using archives in PSO for MO problems will improve their performance directly. Recently, Parsopoulos and Tasoulis proposed a vector evaluated particle swarm optimization (VEPSO) [19] which adopted a ring migration topology and PVE system to simultaneously work 2 to 10 CPUs to find non–dominated solutions. Coello Coello et al. proposed a MOPSO method which incorporates Pareto dominance and a special mutation operator to solve multi-objective optimization problems [20].
3135 c 1-4244-1340-0/07$25.00 2007 IEEE
Although there are numerous approaches of MOPSO, premature convergence, diversity and solutions located on/closer to the Pareto front when solving MO problems are still the main deficiency. In this paper, a local guide distribution method called cross-searching strategy (CSS) is proposed for multi-objective particle swarm optimization. The CSS will pickup two suitable local guides for each particle of the population. Through the CSS, the searching ability of each particle will more capable to find more solutions located on/near to the Pareto front. The rest of the paper is organized as follows: in Section II the basic MO concepts are described, Section III describes the proposed method, Section IV presents the experimental results and Section V of the paper contains the conclusion. II. BASIC CONCEPTS Due to the multi-criteria nature of MO problems, the “optimality” of a solution has to be redefined, giving rise to the concept of Pareto optimality. In contrast to the single–objective optimization case, MO problems are characterized by trade–offs and thus, a multitude of Pareto optimal solutions. In a minimization problem, Pareto dominance and Pareto optimality are defined as follows: Definition 1 (Pareto Dominance): A solution x(1) is said to dominate the other solution x(2), if both statement below are satisfied. (1) (2) z The solution x is no worse than x in all objectives, or fj(x(1))< fj(x(2)) for all j=1,2,…,M (1) (2) z The solution x is strictly better than x in at least one (1) (2) for at least objective, or fj(x )
