空间科学与技术学院学术论文集西安电子科技大学Word下载.docx

1、空间科学与技术学院学术论文集西安电子科技大学#A Decomposition Based Multi-objective Particle Swarm Optimizer for Constraint Handling#Weikang Ning, Baolong Guo, Yunyi Yan, Jinfu Wu, Dan Zhao#（School of Aerospace Science &# Technology, XidianUniv., Xian 710071, China;#）#Abstract:# Multi-objective particle swarm optimization

2、 （MOPSO） algorithms based on decomposition have drawn a lot attention recently. Despite the success of decomposition based MOPSO （MOPSO/D） algorithms, its use in constrained multi-objective optimization problems （CMOPs） remains to be further studied. Most MOPSO/D algorithms proposed recently are des

3、igned for unconstrained problems. Thus we aim to extend the ability of MOPSO/D algorithms in our study. A framework of decomposition based MOPSO for constraint handling （cMOPSO/D） is first proposed in this paper. Then two typical constraint handling techniques are combined with cMOPSO/D as a compari

4、son, which results in two versions of cMOPSO/D:# cMOPSO/D using penalty function （cMOPSO/D-P） and cMOPSO/D using lexicographic ordering （cMOPSO/D-LO）. The algorithms proposed are tested on ten CMOPs and are compared with three state-of-the-art algorithms, cMOEA/D, D2MOPSO and OMOPSO. Experimental re

5、sults supported by the statistical analysis of three quantitative metrics, together with some theoretical analysis, suggest that the proposed algorithms are effective, competitive and promising.#Keywords:# Multi-objective optimization, Decomposition, PSO, Constraint handling, Penalty function, Lexic

6、ographic ordering#1. Introduction#As a paradigm of evolutionary algorithm （EA）, particle swarm optimization （PSO） has been successfully applied to solving multi-objective optimization problems （MOPs）22,20,11. In the real world, however, many problems to be solved often need to satisfy several equali

7、ty and/or inequality constraints, and this presents an additional challenge for multi-objective evolutionary algorithms （MOEAs）. Solving constrained multi-objective problems （CMOPs） ideally often requires the search making a balance between the feasible and infeasible regions, thus the information c

8、arried by the infeasible solutions could be fully exploited.#Many constraint handling techniques are proposed and have been studied for a long time in single objective optimization7,13. Penalty function is widely used for its simplicity 28,18,33,2. The fitness of a solution is calculated based on it

9、s objective value and a penalty term. But how to determine the amount of penalization is rather hard. To alleviate this drawback, some adaptive penalty functions are reported in the literature 17,8,3,5. Besides, different constraint handling techniques based on lexicographic ordering are also widely

10、 studied. Deb9 proposed a binary tournament selection operator to compare two solutions. Based on the satisfaction level for the constraints, Takahama and Sakai proposed an constrained method #29 that uses level comparison to compare two solutions. As an improvement of con-strained method, constrain

11、ed method, which was proposed by Takahama and Sakai 30, adopts an level comparison to#compare two solutions. A dynamic control of was also proposed and the comparison scheme was combined with PSO. Because it is hard to strike the right balance between the objective andthe penalty term in penalty fun

12、ctions, Runarsson and Yao23 proposed a statistic ranking method in which the constraint violation is ignored with some probability. #In relatively traditional Pareto dominance based MOEAs, constraint-dominance is a widely used constraint handling technique. Inspired by the binary tournament selectio

13、n operator 9 used in constrained single objective optimization, a constraint-dominance relationship was proposed by Deb 10 in NSGA-II to compare two solutions. Similar constraint handling strategies are adopted by MOPSO 6 and OMOPSO 27. Besides, penalty function based constraint handling techniques

14、are also used. An adaptive penalty based constrained handling technique was incorporated with non-dominated sorting by Woldesenbet 32. In this method, each objective value of a solution is modified first based on its constraint violation and then non-dominated sorting is used.#The concept of decompo

15、sition, which was proposed by Zhang37 recently, has proven to be efficient in handling many complex MOPs. The fitness assignment mechanism adopted by multi-objective evolutionary algorithm based on decomposition （MOEA/D） makes it easier for MOEA/D to incorporate many constraint handling techniques o

16、riginally invented for single objective optimization. Jan and Zhang14 proposed a penalty function based MOEA/D for constraint handling. The penalty function adopted uses an adaptive threshold to control how heavy an infeasible solution is punished. Besides, two lexicographic ordering based MOEA/D are also reported. Asafuddoula4 introduced an allowable violation threshold to determine if an infeasible solution is considered as feasible. The thre