Differential evolution powered by collective information
Introduction
Differential evolution (DE) is a simple but powerful evolutionary algorithm (EA) proposed by Storn and Price [32]. Similar to other EAs, it utilizes a population-based stochastic search method instead of complex mathematical operations. Over the past two decades, DE has received much attention from diverse domains of scientific and engineering research. Not only have other advanced variants [6], [7], [15], [26], [28], [31], [38], [43] evolved for different optimization problems, but DE has also been successfully applied to various real-world problems, such as electromagnetics [29], optics [49], pattern recognition [37] and signal processing [2].
In DE, the key effort to explore new components for potential solutions relies on the mutation process. It facilitates the search by providing a mutant vector, which is used in the next process (called crossover) for the generation of solution candidates. In most mutation strategies, the ingredients of the mutant vector are based on various randomly picked vector solutions or the best one in the current population. In some advanced DE variants [16], [47], [48], the choice of vector may extend to one from a group of top ranked vectors or the best group from a randomly selected subset of the population.
As reflected by those advanced DE variants, a high-quality vector (in terms of fitness) is important, as it provides a preferable direction for the search. It is also common for only a single vector of this type to be employed. This is different from what is witnessed in other engineering aspects, such as human swarming [24] and IQ Social (IQS) [34], in which a significant impact of collective intelligence is observed. It is thus our motivation to study how collective information can impact DE design.
Alternately, DE suffers from the problem of stagnation, which deteriorates its performance [17]. When stagnation occurs, no better solution can be found from the newly created candidate solutions, even though the population remains diverse. As noted in [17], the occurrence probability of stagnation depends on the number of different potential trial vectors available and their chances of survival in the following generations.
In this paper, we propose a novel mutation strategy (CIM) that takes advantage of the collective information from the m best vectors in the population, where m (m∈ [1, i]) is a random integer governed by the target vector of rank i. A combinational vector is formed by weighted contributions from these m best vectors and then used as a component to generate a mutant vector. This combinational vector is useful not only in mutation but also in crossover. We thus design a collective information-based crossover (CIX) that can help to relieve the problem of stagnation. Based on the proposed CIM and CIX operators, a collective information-powered DE variant named CIPDE is proposed. Detailed analyses are performed by evaluating its performance on 28 benchmark functions developed for the 2013 IEEE Congress on Evolutionary Computation (IEEE CEC2013) [19].
Our contributions are multifold. We are the first to propose the use of collective information from multiple best vectors in mutation and crossover. Moreover, based on the proposed mutation and crossover operations, we design a new DE variant, which achieves a better solution, as confirmed with the use of the CEC2013 test suite.
The rest of this paper is organized as follows. In Section 2, an overview of DE algorithms and an introduction to collective intelligence are presented. In Section 3, the proposed operations, CIM and CIX, together with the new DE variant, CIPDE, are described in detail. Extensive experiments were carried out; the results are presented in Section 4, where detailed comparison and analyses are also given. Finally, the study's conclusions are given in Section 5.
Section snippets
Basis
DE provides a direct search method for continuous optimization problems. It consists of four basic components: initialization, mutation, crossover and selection.
Initialization: Consider a D-dimensional problem; an initial population is randomly generated within the search domain, where NP is the population size. Moreover, other user-defined operational parameters are specified.
After the initialization, DE enters a simple loop of mutation,
Motivation
As described in Section 2.2, the vectors involved in most mutation strategies are either randomly selected or the best in the population. However, multiple high-performing vectors are considered in some approaches, such as JADE [48], in which (7) is proposed, and they still rely on a single good solution.
Here, we explore the use of collective information from top performers in the population to improve DE. To express the design motivation, a simple illustrative example is given, as follows:
Experimental study
In this section, the effectiveness of the proposed CIM and CIX operators and the superiority of the presented CIPDE algorithm are verified by comprehensive experiments conducted on the CEC2013 benchmark function set [19]. The CEC2013 set includes a wide range of optimization functions, such as unimodal functions (F1−F5), basic multimodal functions (F6−F20) and composition functions (F21−F28).
The performance of different algorithms is evaluated based on the solution error value, which is defined
Conclusions
In this paper, a DE variant called CIPDE that is powered by collective information is proposed. A collective information-based mutation (CIM) operator and a collective information-based crossover (CIX) operator are presented. CIM creates promising potential solutions by utilizing the normalized linear combination of the m best vectors that are better than the target vectors in terms of fitness values. CIX is designed to prevent stagnation by efficiently utilizing the collective information to
Acknowledgments
The work described in this paper was supported in part by the National Natural Science Foundation of China (No. 61401523), in part by the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (No. 2014KQNCX002), in part by the International Science & Technology Cooperation Program of China (No. 2015DFR11050), and in part by the External Cooperation Program of Guangdong Province of China (No. 2013B051000060).
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