Elsevier

Neurocomputing

Volume 290, 17 May 2018, Pages 130-147
Neurocomputing

An improved multi-population ensemble differential evolution

https://doi.org/10.1016/j.neucom.2018.02.038Get rights and content

Abstract

Differential evolution (DE) is a population-based stochastic optimization technique that can be applied to solve global optimization problems. The selected mutation strategies and the control parameters can affect the performance of DE. As mutation strategies have significant effects on solving optimization problems, multiple mutation strategies of DE have been developed. Multi-population ensemble DE (MPEDE) realizes an ensemble of multiple strategies, but “DE/rand/1” may be slow at exploitation of the solutions, and the control parameter by applying arithmetic mean in “DE/current-to-pbest/1” may cause premature convergence. Address these issues, an improved multi-population ensemble DE (IMPEDE) is proposed in this paper. IMPEDE proposes a new mutation strategy “DE/pbad-to-pbest/1” instead of the mutation strategy “DE/rand/1” in MPEDE, and the new strategy utilize not only the good solution information(pbest), but also the information of the bad solution (pbad) toward the good solution to balance exploration and exploitation. Furthermore, IMPEDE employs the improved parameter adaptation approach to avoid premature convergence of the “DE/current-to-pbest/1” strategy by adding the weighted Lehmer mean strategy. Experiments have been conducted with CEC2005 and CEC2017 benchmark functions, and the results have demonstrated that IMPEDE outperforms than that of MPEDE and the other four recent proposed DE methods in obtaining the global optimum and accelerating the convergence speed.

Introduction

Differential evolution (DE), which was firstly proposed by Storn and Price, is population-based, stochastic optimization algorithm [1], [2]. Also, DE is a simple and efficient evolutionary algorithm (EA) for global optimization that has been successfully applied to various optimizati on problems [3], [4], [5], [6], [7], [8]. Despite of its simplicity, DE has been shown to be competitive with other EAs and applies mutation, crossover, and selection operations at each generation to guide its population toward the global optimum [9], [10], [11], [12].

However, two crucial components may significantly influence the optimization performance of DE. One is mutation strategy, and the other is control parameters such as population size NP, mutation factor F, and the crossover probability CR [13], [14]. There are many researchers paying considerable attentions to choosing the suitable mutation strategy and control parameters [15], [16], [17], [18]. In addition, these experiences are immensely helpful in enhancing the performance of DE.

Zhang and Sanderson [19] have proposed a well-known, effective DE variant Adaptive Differential Evolution with Optional External Archive (JADE) which uses a novel mutation strategy called “DE/current-to-pbest/1” with an archive and also employs a control parameter adaptation mechanism. The “DE/current-to-pbest/1” is a mutation strategy which directs the generation of mutant vector towards the best and the other good member of the population. The archive is utilized to add the parent solutions, when the parent solutions failed in the selection process. The “DE/current-to-pbest/1” with an archive is not liable to be trapped into the local optimum and very useful in solving complex optimization problems such as the unimodal problem and the multimodal problem. Motivated by JADE, Wu et al. [20] have introduced multi-population ensemble DE(MPEDE), in which a multi-population based approach is utilized to realize a dynamic ensemble of multiple mutation strategies consisting of “DE/current-to-pbest/1” with an archive, “DE/current-to-rand/1”, and “DE/rand/1”. “DE/rand/1” is a mutation strategy which directs the generation of mutant vector towards the random member of the population. In addition, “DE/current-to-rand/1”, a rotation-invariant strategy without the crossover operation, is extremely effective in solving rotated problems. Moreover, control parameters of MPEDE are adapted based on the mechanism of JADE. However, there are some issues in these works. “DE/rand/1” mutation strategy may be good at exploring the search space, but may be slow at exploitation of the solutions [21], [22]. In the control parameter adaptation mechanism of “DE/current-to-pbest/1”, the control parameter by applying arithmetic mean has an implicit bias to small values during self-adaptation process and causes premature convergence at the end [23], [24]. Therefore, by integrating the advantages and overcoming the disadvantages of JADE and MPEDE, we have proposed an improved multi-population ensemble differential evolution algorithm (IMPEDE).

IMPEDE proposes a new mutation strategy “DE/pbad-to-pbest/1” to balance exploration and exploitation with the objective of obtaining optimal solution and accelerating convergent speed instead of the “DE/rand/1” mutation strategy in MPEDE. Meanwhile, because the “DE/current-to-pbest/1” with an archive and “DE/current-to-rand/1” are the key strategies for solving optimization problems in MPEDE, we combine these two strategies with a new mutation strategy “DE/pbad-to-pbest/1” called the improved multi-population based mutation strategy ensemble approach in IMPEDE to search the global optimal solution using MPEDE framework. Furthermore, to tackle the issue of premature convergence caused by applying arithmetic mean in the control parameter of “DE/current-to-pbest/1”, IMPEDE employs the improved parameter adaptation approach to make slight modifications on the “DE/current-to-pbest/1” strategy by adding the weighted Lehmer mean strategy on the adaptation of control parameter. Experimental results have shown that the IMPEDE is more efficient than that of the previous DE algorithms. It can achieve the goals of accelerating the speed of convergence and jumping out of local optimum.

The rest of the paper is organized as follow: Section 2 introduces the original DE and its development briefly. The IMPEDE is proposed in Section 3. In Section 4, the performance of the proposed IMPEDE algorithm is evaluated using the CEC2005 and CEC2017 benchmark functions and compared with other DE algorithms. IMPEDE is also applied to solve a real-life problem. Finally, the research limitation and future works are discussed in Section 5 and conclusions are given in Section 5.

Section snippets

Related work

Differential evolution is a parallel direct search method that encodes the candidate solution, i.e.Xi,G={xi,G1,,xi,GD},i=1,,NP, where D is the dimension of the problem and NP is the population size. DE enters an evolutionary process which contains mutation, crossover, and selection.

During the mutation process, the mutant vector Vi,G is generated by employing one of the following mutation strategies, corresponding to each member or the target vector Xi,G.

DE/rand/1: Vi,G=Xr1,G+F·(Xr2,GXr3,G)

Overview

IMPEDE adopts the improved Multi-population based mutation strategy ensemble approach which combines mutation strategies of “DE/current-to-pbest/1” with an archive and “DE/current-to-rand/1”, and “DE/pbad-to-pbest/1”, a new mutation strategy of which we propose instead of the mutation strategy “DE/rand/1” in MPEDE algorithm. Due to less efficient of “DE/rand/1” in terms of convergence rate, we propose a new mutation strategy “DE/pbad-to-pbest/1” which utilizes not only the good solution

Experimental parameters setting

The performance of the proposed algorithm is evaluated and compared with the other DE algorithms such as JADE [19]; jDE [33]; SaDE [15]; CoDE [34] and MPEDE [20]. JADE implements a new mutation strategy “DE/current-to-pbest” with optional external archive and updates control parameters in an adaptive manner. The jDE algorithm employs a self-adaptation scheme for the DE control parameters. SaDE utilizes the self-adaptive mutation strategies and respective control parameters based on their

Conclusion and future work

In this paper, we have proposed an improved Multi-population ensemble differential evolution algorithm (IMPEDE). Compare to the previous work, IMPEDE has shown the strong capability of jumping out of the local optimum and fasten the speed of the convergence. Because the “DE/current-to-pbest/1” with an archive and “DE/current-to-rand/1” are the key strategies for solving optimization problems in MPEDE, IMPEDE combine these two strategies with a new mutation strategy “DE/pbad-to-pbest/1” called

Acknowledgments

The authors would like to express their sincere thanks to the Associate Editor and the anonymous reviewers for their valuable suggesti ons and comments on this paper. This work is supported by the National Natural Science Foundation of China (61563012, 61203109), Guangxi Natural Science Foundation (2014GXNSFAA118371, 2015GXNSFBA139260).

Lyuyang Tong was born in Hubei, China, in 1992. He received the B.E. degree in Computer Science and Technology and also the B.A. degree in Japanese Language from Changchun University of Technology, China, in 2015 and 2017, respectively. Now he is pursuing the M.S. degree in Software engineering from Guilin University of Technology, China. His current research interests include evolutionary computation, multi-objective evolutionary optimization and machine learning.

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    Lyuyang Tong was born in Hubei, China, in 1992. He received the B.E. degree in Computer Science and Technology and also the B.A. degree in Japanese Language from Changchun University of Technology, China, in 2015 and 2017, respectively. Now he is pursuing the M.S. degree in Software engineering from Guilin University of Technology, China. His current research interests include evolutionary computation, multi-objective evolutionary optimization and machine learning.

    Minggang Dong received the B.S. degree from Naval University of Engineering, China, in 2000, the M.S. degree from Guangxi University, China, in 2006, and the Ph.D. degree from Zhejiang University, China, in 2012. He is currently a Professor in the College of Information Science and Engineering at Guilin University of Technology, China. His current research interests include machine learning, evolutionary computation, parallel and distributed computation.

    Chao Jing received his Ph.D. degree in Computer Science from Shanghai Jiao Tong Univeristy, China 2014. From July 2014, he is an Assistant Professor with School of Information Science and Engineering at Guilin University of Technology. Since Sept 2017, he has been worked as a visiting scholar at the School of Engineering, Brown Univeristy. His research interests include Big Data & Cloud Computing, Energy-efficient Computing and Optimzation scheduling techniques.

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