Performance of global random search algorithms for large dimensions
We investigate the rate of convergence of general global random search (GRS) algorithms. We show that if the dimension of the feasible domain is large then it is impossible to give any guarantee that the global minimizer is found by a general GRS algorithm with reasonable accuracy. We then study precision of statistical estimates of the global minimum in the case of large dimensions. We show that these estimates also suffer the curse of dimensionality. Finally, we demonstrate that the use of quasi-random points in place of the random ones does not give any visible advantage in large dimensions.
KeywordsGlobal optimization Statistical models Extreme value statistics Random search
The work of the first author was partially supported by the SPbSU Project No. 6.38.435.2015 and the RFFI Project No. 17-01-00161. The work of the second author was supported by the Russian Science Foundation, Project No. 15-11-30022 ‘Global optimization, supercomputing computations, and applications’. The work of the third author was supported by the Research Council of Lithuania under Grant No. MIP-051/2014.
- 1.Auger, A., Hansen, N.: Theory of evolution strategies: a new perspective. In: Auger, A., Doerr, B. (eds.) Theory of Randomized Search Heuristics: Foundations and Recent Developments, pp. 289–325. World Scientific Publishing, Singapore (2010)Google Scholar
- 6.Nevzorov, V.B.: Records: mathematical theory. American Mathematical Soc. (2001)Google Scholar
- 7.Niederreiter, H.: Random number generation and quasi-monte carlo methods, cbms-nsf reg. In: Conference of series applied mathematics, vol. 63 (1992)Google Scholar
- 10.Zhigljavsky, A.: Monte-Carlo methods in global optimization, PhD thesis. Leningrad University (1981)Google Scholar
- 11.Zhigljavsky, A.: Mathematical Theory of Global Random Search. Leningrad University Press, Leningrad (1985). (in Russian)Google Scholar
- 15.Zhigljavsky, A., Žilinskas, A.: Methods of Seeking a Global Extremum. Nauka, Moscow (1991)Google Scholar