Enhancing the search ability of differential evolution through orthogonal crossover
Introduction
Differential evolution (DE), proposed by Storn and Price in 1995 [36], [37], is a class of simple yet efficient evolutionary algorithms (EAs) for continuous optimization problems. It has been successfully used in various applications (e.g., [2], [8], [21], [30], [46]). Like other EAs, DE is a population-based stochastic optimization method. It adopts mutation and crossover operators to search for new promising areas in the search space. The commonly used crossover operators in DE are binomial crossover and exponential crossover. Note, however, that these two crossover operators can only generate one new solution, which is a vertex of a hyper-rectangle defined by two parent solutions. This could somehow limit the search ability of DE.
Observing that the reproduction of new solutions in EAs can be considered as “experiments”, Zhang and his co-workers [50], [51] used experimental design methods to design genetic operators and proposed orthogonal crossover (OX). OX operators can make a systematic and statistically sound search in a region defined by parent solutions. OX operators have been successfully applied in various optimization problems. Leung and Wang [24] introduced a quantization technique to OX for dealing with numerical optimization. Experimental studies show that their quantization OX (QOX) operator is effective and efficient in a number of numerical optimization test instances.
The main purposes of this paper are twofold: (1) to reveal the limitation of the commonly used crossover operators in DE, and (2) to verify that the search ability of DE can be enhanced by effectively probing the hyper-rectangle defined by the mutant and target vectors. To achieve the second purpose, we have suggested a framework for using QOX in DE variants and presented an OX-based DE (OXDE), which is a combination of DE/rand/1/bin and QOX, as an instantiation of our framework. OXDE is simple and easy to implement. It uses QOX to complement binomial crossover or exponential crossover for searching some promising regions in the solutions space. Experimental results indicate that QOX significantly improves the performance of DE/rand/1/bin. Some effort has also been made not to increase the number of control parameters in our framework. Moreover, we also show that our framework can be used for improving the performance of several other DE variants.
The remainder of this paper is organized as follows. Section 2 introduces DE and Section 3 reviews the related work. In Section 4, the idea of OX operators is briefly explained. Our framework and an implementation, that is, OXDE, are presented in Section 5. In Section 6, extensive experiments have been carried out to test OXDE and to study the effectiveness of our framework on several other DE variants. Some discussions on OXDE have also been provided in this section. Finally, we conclude this paper in Section 7.
Section snippets
Differential evolution
DE is for solving the following continuous global optimization problem:where is continuous and ∀i ∈ {1, … , D}, −∞ < ai < bi < +∞.
DE maintains a population of NP individual members, where NP is the population size, and each member is a point in the solution space S. DE improves its population generation by generation. It extracts distance and direction information from the current population for generating new solutions for the next generation. Almost all the
The related work
Much effort has been made to improve the performance of DE and a number of DE variants have been proposed. These improvements can be classified into four categories:
- (1)
Dynamic adaptation or self-adaptation of the control parameters (i.e., NP, F, and CR) in DE to ease the task of control parameter setting and to dynamically or self-adaptively adjust the search behavior of DE to suit different landscapes [1], [6], [9], [23], [26], [32], [33], [41], [47], [52]. For example, Lee et al. [23] estimated
Orthogonal design
Consider a system whose cost depends on K factors (i.e., variables), each factor can take one of Q levels (i.e., values). To find the best level for each factor to minimize the system cost, one can do one experiment for every combination of factor levels and then select the best one if K and Q are small. The number of all the combinations is QK. Therefore, it is not possible or efficient to test all the combinations in the case when K and Q are large. Experimental design methods can be used for
Basic idea
Crossover operators play a key role in DE. As shown in Fig. 1, both binomial and exponential crossover operators, the two most commonly used crossover operators in DE, only generate and evaluate one single trial vector , which is a vertex of the hyper-rectangle defined by the mutant vector and the target vector . Thus, they do not carry out a systematic search in this hyper-rectangle, which might be a promising region in the solution space. Therefore, the search ability of DE
Experimental settings
A suite of 24 test instances is used for our experimental studies. The first 10 test instances are widely used in the evolutionary computation community [31], and the other 14 test instances are the first 14 test instances designed for the CEC2005 Special Session on real-parameter optimization [38].
Since OXDE is presented to enhance the search ability of DE/rand/1/bin, the performance comparison is mainly done between DE/rand/1/bin and OXDE. The orthogonal array used in OXDE is L9(34). The
Conclusion
The commonly used crossover operators in current popular DE only visit one vertex of the hyper-rectangle defined by the mutant and target vectors, this could confine the algorithm search ability. In this paper, we have attempted to introduce QOX into DE for overcoming this shortcoming. QOX is able to make a systematic and rational search in the region defined by the two parent solutions. We have suggested a framework which uses QOX in DE and proposed a hybrid algorithm of DE/rand/1/bin with
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 60805027, 90820302, and 61175064), and in part by the Research Fund for the Doctoral Program of Higher Education of China (Grant No. 200805330005). The authors thank the anonymous reviewers for their very helpful and constructive comments and suggestions.
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