Article

Full entry | PDF   (0.5 MB)
Keywords:
optimization; cost function; global minimum; global convergence; local convergence; differential evolution algorithm; optimal solution set; convergence in probability; numerical testing
Summary:
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.
References:
[1] Hu, Z., Xiong, S., Su, Q., Zhang, X.: Sufficient conditions for global convergence of differential evolution algorithm. J. Appl. Math. 2013 (2013), Article ID 139196, 14 pages. DOI 10.1155/2013/193196 | MR 3122108
[2] Mlýnek, J., Knobloch, R., Srb, R.: Mathematical model of the metal mould surface temperature optimization. AIP Conference Proceedings 1690 AIP Publishing, Melville (2015), Article No. 020018, 8 pages. DOI 10.1063/1.4936696
[3] Mlýnek, J., Knobloch, R., Srb, R.: Optimization of a heat radiation intensity and temperature field on the mould surface. ECMS 2016 Proceedings, 30th European Conference on Modelling and Simulation Regensburg, Germany (2016), 425-431. DOI 10.7148/2016-0425 | MR 3203813
[4] Price, K. V.: Differential evolution: a fast and simple numerical optimizer. Proceedings of North American Fuzzy Information Processing Berkeley (1996), 524-527. DOI 10.1109/NAFIPS.1996.534790
[5] Price, K. V., Storn, R. M., Lampien, J. A.: Differential Evolution. A Practical Approach to Global Optimization. Natural Computing Series. Springer, Berlin (2005). DOI 10.1007/3-540-31306-0 | MR 2191377 | Zbl 1186.90004
[6] Simon, D.: Evolutionary Optimization Algorithms. Biologically Inspired and Population-Based Approaches to Computer Intelligence. John Wiley & Sons, Hoboken (2013). MR 3362741 | Zbl 1280.68008
[7] Storn, R. M., Price, K. V.: Differential evolution---a simple and efficient heuristics for global optimization over continuous spaces. J. Glob. Optim. 11 (1997), 341-359. DOI 10.1023/A:1008202821328 | MR 1479553 | Zbl 0888.90135

Partner of