Utilizing cumulative population distribution information in differential evolution

doi:10.1016/j.asoc.2016.07.012

Applied Soft Computing

Volume 48, November 2016, Pages 329-346

https://doi.org/10.1016/j.asoc.2016.07.012 Get rights and content

Highlights

•
Due to the fact that single population fails to contain enough information to estimate the covariance matrix reliably, this paper updates the covariance matrix in DE by an adaptation procedure, which makes use of the cumulative distribution information of the population.
•
CPI-DE provides a simple yet efficient synergy of two kinds of crossover: the crossover in the Eigen coordinate system and the crossover in the original coordinate system. The former aims at identifying the properties of the fitness landscape and improving the efficiency and effectiveness of DE by producing the offspring toward promising directions. In addition, the purpose of the latter is to maintain the superiority of the original DE. Moreover, no extra parameters are required in CPI-DE.
•
Our experimental studies have shown that CPI-DE is capable of enhancing the performance of several classic DE versions and advanced DE variants.

Abstract

Differential evolution (DE) is one of the most popular paradigms of evolutionary algorithms. In general, DE does not exploit distribution information provided by the population and, as a result, its search performance is limited. In this paper, cumulative population distribution information of DE has been utilized to establish an Eigen coordinate system by making use of covariance matrix adaptation. The crossover operator of DE implemented in the Eigen coordinate system has the capability to identify the features of the fitness landscape. Furthermore, we propose a cumulative population distribution information based DE framework called CPI-DE. In CPI-DE, for each target vector, two trial vectors are generated based on both the original coordinate system and the Eigen coordinate system. Then, the target vector is compared with these two trial vectors and the best one will survive into the next generation. CPI-DE has been applied to two classic versions of DE and three state-of-the-art variants of DE for solving two sets of benchmark test functions, namely, 28 test functions with 30 and 50 dimensions at the 2013 IEEE Congress on Evolutionary Computation, and 30 test functions with 30 and 50 dimensions at the 2014 IEEE Congress on Evolutionary Computation. The experimental results suggest that CPI-DE is an effective framework to enhance the performance of DE.

Graphical abstract

The mutation, crossover, and selection of CPI-DE.

Introduction

Differential Evolution (DE), proposed by Storn and Price [1], [2] in 1995, is a very popular evolutionary algorithm (EA) paradigm. During the past two decades, DE has attracted a lot of attention and has been successfully applied to solve a variety of numerical and real-world optimization problems [3], [4], [5].

The remarkable advantages of DE are its simple structure and ease of implementation. In DE, each individual in the population is called a target vector. DE contains three basic operators: mutation, crossover and selection. During the evolution, DE generates a trial vector for each target vector through the mutant and crossover operators. Afterward, the trial vector competes with its target vector for survival according to their fitness. DE also involves three control parameters: the population size, the scaling factor, and the crossover control parameter. The performance of DE is dependent mainly on these three operators and three control parameters. In order to further improve the performance of DE, a lot of DE variants have been designed, such as JADE [6], jDE [7], SaDE [8], EPSDE [9], CoDE [10], and so on.

DE is a population-based optimization algorithm; however, population distribution information has not yet been widely utilized in the DE community, which makes DE inefficient especially when solving some optimization problems with complex characteristics. Very recently, two attempts have been made along this line [11], [12]. However, the methods proposed in Refs. [11], [12] only utilize the distribution information from a single population of one generation, and the cumulative distribution information of the population over the course of evolution has been ignored. Moreover, these methods introduce some extra parameters. Therefore, new insights into the usage of the population distribution information in DE are quite necessary.

In 2001, Hansen and Ostermeier [13] proposed the well-known covariance matrix adaptation evolution strategy, called CMA-ES. CMA-ES generates offspring by sampling a multivariate normal distribution, which includes three main elements: mean vector of the search distribution, covariance matrix, and step-size. Indeed, covariance matrix reflects the population distribution information to a certain degree [12]. In CMA-ES, the covariance matrix is self-adaptively updated according to the information from the previous and current generations.

In this paper, we make use of the cumulative distribution information of the population to establish an Eigen coordinate system in DE, by considering CMA as an effective tool. Furthermore, we suggest a cumulative population distribution information based DE framework called CPI-DE. In CPI-DE, for each target vector, the crossover operator of DE is implemented in both the original coordinate system and the Eigen coordinate system and, as a result, two trial vectors are generated. Subsequently, the target vector is compared with these two trial vectors and the best one will enter the next population. CPI-DE is applied to two classic DE versions as well as three state-of-the-art DE variants. Extensive experiments across two benchmark test sets from the 2013 IEEE Congress on Evolutionary Computation (IEEE CEC2013) [14] and the 2014 IEEE Congress on Evolutionary Computation (IEEE CEC2014) [15] have been implemented to verify the effectiveness of CPI-DE.

The main contributions of this paper can be summarized as follows:

•
Due to the fact that single population fails to contain enough information to estimate the covariance matrix reliably, this paper updates the covariance matrix in DE by an adaptation procedure, which makes use of the cumulative distribution information of the population.
•
CPI-DE provides a simple yet efficient synergy of two kinds of crossover: the crossover in the Eigen coordinate system and the crossover in the original coordinate system. The former aims at identifying the properties of the fitness landscape and improving the efficiency and effectiveness of DE by producing the offspring toward the promising directions. In addition, the purpose of the latter is to maintain the superiority of the original DE. Moreover, no extra parameters are required in CPI-DE.
•
Our experimental studies have shown that CPI-DE is capable of enhancing the performance of several classic DE versions and advanced DE variants.

The rest of this paper is organized as follows. Section 2 describes the basic procedure of DE. Section 3 briefly reviews the recent developments of DE in the last five years. The proposed CPI-DE is presented in Section 4. The experimental results and the performance comparison are given in Section 5. Finally, Section 6 concludes this paper.

Section snippets

Differential evolution (DE)

Similar to other EA paradigms, DE starts with a population of NP individuals, i.e., $P^{(g)} = {{\vec{x}}_{i}^{(g)} = (x_{i, 1}^{(g)}, ..., x_{i, D}^{(g)}), i = 1,..., N P}$ , where g is the generation number, D is the dimension of the decision space, and NP is the population size. In $P^{(g)}$ , each individual is also called a target vector. At g = 0, the jth decision variable of the ith target vector is initialized as follows: $x_{i, j}^{(0)} = L_{j} + r a n d (0,1) * (U_{j} - L_{j}), i = 1,..., N P, j = 1,..., D$ where rand(0,1) represents a uniformly distributed random number

The related work

Recent two decades have witnessed significant progress in the developments of DE. In 2011, Das and Suganthan [16] presented a comprehensive survey on DE, including the basic concepts and major variants of DE, as well as the applications and theoretical studies of DE. Next, we will briefly introduce the recent developments of DE in the last five years.

Motivation

Based on the above introduction, it is clear that population distribution information has seldom been involved in the current state-of-the-art DE.

Very recently, Guo and Yang [11] and Wang et al. [12] made the first attempt to exploit the population distribution information in DE. The methods proposed in Refs. [11], [12] share some similar ideas. More specifically, these two methods firstly compute the covariance matrix of the population. Subsequently, the Eigenvectors obtained from the Eigen

Experimental study

In this paper, two sets of benchmark test functions are employed to demonstrate the effectiveness of CPI-DE, i.e., 28 test functions with 30 dimensions (30D) and 50 dimensions (50D) at IEEE CEC2013 [14], and 30 test functions with 30 dimensions (30D) and 50 dimensions (50D) at IEEE CEC2014 [15]. The 28 test functions in the first set are denoted as CEC2013₁-CEC2013₂₈, and the 30 test functions in the second set are denoted as CEC2014₁-CEC2014₃₀.

In our experiments, the function error value $(f ({\vec{x}}^{b}$

Conclusion

A simple yet efficient DE framework, which is referred as CPI-DE, has been presented in this paper. In CPI-DE, the cumulative population distribution information is utilized to establish an Eigen coordinate system for DE’s crossover. Moreover, CPI-DE performs the crossover in both the original coordinate system and the Eigen coordinate system in a deterministic manner. As a result, two trial vectors are generated for each target vector and the best one among the target vector and two trial

Acknowledgments

The authors would like to thank the anonymous reviewers for their very constructive and helpful suggestions. This work was supported in part by the National Basic Research Program 973 of China (Grant No. 2011CB013104), in part by the Innovation-driven Plan in Central South University (No. 2015CXS012 and No. 2015CX007), in part by the National Natural Science Foundation of China under Grant 61273314, in part by the EU Horizon 2020 Marie Sklodowska-Curie Individual Fellowships (Project ID:

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View full text

Utilizing cumulative population distribution information in differential evolution

Highlights

Abstract

Graphical abstract

Introduction

Section snippets

Differential evolution (DE)

The related work

Motivation

Experimental study

Conclusion

Acknowledgments

Inf. Sci.

Appl. Soft Comput.

Appl. Soft Comput.

Appl. Soft Comput.

Appl. Soft Comput.

Math. Probl. Eng.

Swarm Evol. Comput.

Inf. Sci.

Appl. Soft Comput.

Inf. Sci.

Expert Syst. Appl.

Inf. Sci.

Neurocomputing

Appl. Math. Comput.

Comput. Oper. Res.

Swarm Evol. Comput.

Inf. Sci.

Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces

J. Global Optim.

MOMMOP: multiobjective optimization for locating multiple optimal solutions of multimodal optimization problems

IEEE Trans. Cybern.

JADE: adaptive differential evolution with optional external archive

IEEE Trans. Evol. Comput.

Self-adapting control parameters in differential evolution: a comparative study on numerical benchmark problems

IEEE Trans. Evol. Comput.

Differential evolution algorithm with strategy adaptation for global numerical optimization

IEEE Trans. Evol. Comput.

Differential evolution with composite trial vector generation strategies and control parameters

IEEE Trans. Evol. Comput.

Enhancing differential evolution utilizing Eigenvector-based crossover operator

IEEE Trans. Evol. Comput.

Completely derandomized self-adaptation in evolution strategies

Evol. Comput.

Problem Definitions and Evaluation Criteria for the CEC 2013 Special Session on Real-parameter Optimization

Problem Definitions and Evaluation Criteria for the CEC 2014 Special Session and Competition on Single Objective Real-parameter Numerical Optimization, Technical Report 201311

Differential evolution: a survey of the state-of-the-art

IEEE Trans. Evol. Comput.

Differential evolution with ranking-based mutation operators

IEEE Trans. Cybern.

Differential evolution with neighborhood and direction information for numerical optimization

IEEE Trans. Cybern.

Differential evolution enhanced with multiobjective sorting-based mutation operators

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