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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 193196, 14 pages
http://dx.doi.org/10.1155/2013/193196
Research Article

Sufficient Conditions for Global Convergence of Differential Evolution Algorithm

1School of Computer Science and Technology, Wuhan University of Technology, Wuhan, Hubei 430070, China
2School of Mathematics and Statistics, Hubei Engineering University, Xiaogan, Hubei 432100, China
3School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China

Received 6 July 2013; Revised 11 August 2013; Accepted 27 August 2013

Academic Editor: Yongkun Li

Copyright © 2013 Zhongbo Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The differential evolution algorithm (DE) is one of the most powerful stochastic real-parameter optimization algorithms. The theoretical studies on DE have gradually attracted the attention of more and more researchers. However, few theoretical researches have been done to deal with the convergence conditions for DE. In this paper, a sufficient condition and a corollary for the convergence of DE to the global optima are derived by using the infinite product. A DE algorithm framework satisfying the convergence conditions is then established. It is also proved that the two common mutation operators satisfy the algorithm framework. Numerical experiments are conducted on two parts. One aims to visualize the process that five convergent DE based on the classical DE algorithms escape from a local optimal set on two low dimensional functions. The other tests the performance of a modified DE algorithm inspired of the convergent algorithm framework on the benchmarks of the CEC2005.