Discrete mathematics and mathematical cybernetics

Generalized heavy-tailed mutation for evolutionary algorithms

A. V. Eremeev^ab, D. V. Silaev^a, V. A. Topchii<sup>ab

a</sup> Novosibirsk State University, 1, Pirogova str., Novosibirsk, 630090, Russia
^b Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia,

Abstract: The heavy-tailed mutation operator, proposed by Doerr, Le, Makhmara, and Nguyen (2017) for evolutionary algorithms, is based on the power-law assumption of mutation rate distribution. Here we generalize the power-law assumption using a regularly varying constraint on the distribution function of mutation rate. In this setting, we generalize the upper bounds on the expected optimization time of the $(1+(\lambda,\lambda))$ genetic algorithm obtained by Antipov, Buzdalov and Doerr (2022) for the OneMax function class parametrized by the problem dimension $n$. In particular, it is shown that, on this function class, the sufficient conditions of Antipov, Buzdalov and Doerr (2022) on the heavy-tailed mutation, ensuring the $O(n)$ optimization time in expectation, may be generalized as well. This optimization time is known to be asymptotically faster than what can be achieved by the $(1+(\lambda,\lambda))$ genetic algorithm with any static mutation rate. A new version of the heavy-tailed mutation operator is proposed, satisfying the generalized conditions, and promising results of computational experiments are presented.

Keywords: Evolutionary algorithms, regularly varying functions, heavy-tailed mutation, optimization time.

UDC: 519.712

MSC: 68Q25;60-08

Received April 15, 2024, published November 1, 2024

DOI: 10.33048/semi.2024.21.062