Towards a stronger theory for permutation-based evolutionary algorithms

Abstract

While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems.To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based (1 + 1) EA proposed by Scharnow, Tinnefeld, and Wegener ( 2004) on the analogues of the L O and J benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation into another one , but also the precise cycle structure of −1 . For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of Θ( ). Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order Θ( ) on jump functions with jump size . CCS CONCEPTS• Theory of computation → Theory of randomized search heuristics.

Runtime Analysis of the SMS-EMOA for Many-Objective Optimization

“…For the bi-objective OJZJ benchmark, again a speed-up of k Ω(k) was proven when optimized via the GSEMO (Zheng and Doerr 2023d) and the NSGA-II (Doerr and Qu 2023a). Several other positive theoretical results exist for this heavy-tailed mutation, or more generally, other heavy-tailed parameter choices (Friedrich, Quinzan, and Wagner 2018;Wu, Qian, and Tang 2018;Quinzan et al 2021;Corus, Oliveto, and Yazdani 2021;Dang et al 2022;Antipov, Buzdalov, and Doerr 2022;Doerr, Ghannane, and Ibn Brahim 2022;Doerr and Rajabi 2023).…”