A First Runtime Analysis of the NSGA-II on a Multimodal Problem
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Very recently, the first mathematical runtime analyses of the multi-objective evolutionary optimizer NSGA-II have been conducted (AAAI 2022, GECCO 2022 (to appear), arxiv 2022). We continue this line of research with a first runtime analysis of this algorithm on a benchmark problem consisting of two multimodal objectives. We prove that if the population size N is at least four times the size of the Pareto front, then the NSGA-II with four different ways to select parents and bit-wise mutation optimizes the OneJumpZeroJump benchmark with jump size 2 ≤ k ≤ n/4 in time O(N n^k). When using fast mutation, a recently proposed heavy-tailed mutation operator, this guarantee improves by a factor of k^Ω(k). Overall, this work shows that the NSGA-II copes with the local optima of the OneJumpZeroJump problem at least as well as the global SEMO algorithm.
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