APPLICATIONS OF MATHEMATICS, Vol. 62, No. 2, pp. 197-208, 2017

The classic differential evolution algorithm and its convergence properties

Roman Knobloch, Jaroslav Mlýnek, Radek Srb

Received September 30, 2016.   First published March 6, 2017.

Roman Knobloch, Jaroslav Mlýnek, Radek Srb, Technical University of Liberec, Studentská 2, 461 17 Liberec, Czech Republic, e-mail: roman.knobloch@tul.cz, jaroslav.mlynek@tul.cz, radek.srb@tul.cz

Abstract: Differential evolution algorithms represent an up to date and efficient way of solving complicated optimization tasks. In this article we concentrate on the ability of the differential evolution algorithms to attain the global minimum of the cost function. We demonstrate that although often declared as a global optimizer the classic differential evolution algorithm does not in general guarantee the convergence to the global minimum. To improve this weakness we design a simple modification of the classic differential evolution algorithm. This modification limits the possible premature convergence to local minima and ensures the asymptotic global convergence. We also introduce concepts that are necessary for the subsequent proof of the asymptotic global convergence of the modified algorithm. We test the classic and modified algorithm by numerical experiments and compare the efficiency of finding the global minimum for both algorithms. The tests confirm that the modified algorithm is significantly more efficient with respect to the global convergence than the classic algorithm.

Keywords: optimization; cost function; global minimum; global convergence; local convergence; differential evolution algorithm; optimal solution set; convergence in probability; numerical testing

Classification (MSC 2010): 60G20, 65K05

DOI: 10.21136/AM.2017.0274-16

Full text available as PDF.


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