A First Runtime Analysis of the NSGA-II on a Multimodal Problem

Qu, Zhongdi; Doerr, Benjamin

doi:10.1007/978-3-031-14721-0_28

Cited by 19 publications

(43 citation statements)

References 33 publications

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“…For this, however, a mild modification of the selection mechanism, proposed earlier [KD06], was needed. In [DQ22], the first mathematical runtime analysis of the NSGA-II on a problem with multimodal objectives was conducted. Again, the NSGA-II was found to be as effective as the (G)SEMO algorithms when the population size was chosen suitably.…”

Section: Previous Workmentioning

confidence: 99%

“….n]}. [DQ22] showed that when using a population of size N ≥ 4(n − 2k + 3) to optimize this benchmark, the NSGA-II algorithm never loses a Pareto-optimal solution value once found. Moreover, O(n k ) iterations are needed in expectation to find the full Pareto front (all Pareto optimal solution values).…”

Section: Benchmark Problemsmentioning

confidence: 99%

“…iterations to find both 1 n and 0 n . To cover any other point on the front, since there is a 0.1 chance that crossover is not applied and since covered points remain covered, the O(N n log n) time for covering the whole inner part of the Pareto front shown in [DQ22] remains valid (and is a lower order term compared to the time to generate the extremal points). In conclusion, in total, with uniform crossover happening at a constant probability, the number of iterations taken is O( N (Cn) k (k−1)!…”

Section: Lemma 3 Consider An Iteration T Of the Nsga-ii Algorithm Opt...mentioning

confidence: 99%

“…Very recently, however, a runtime analysis was conducted [ZLD22] for the NSGA-II, the most common evolutionary multi-objective (EMO) algorithm [DPAM02] (47000 citations on Google scholar). This work was quickly followed up in different directions [ZD22,BQ22,DQ22]. The majority of these studies regards a simplified version of the NSGA-II that does not use crossover.…”

Section: Introductionmentioning

confidence: 99%

“…To this aim, we regard the OneJumpZeroJump benchmark [DZ21], which is a bi-objective version of the classic Jump benchmark intensively studied in the analysis of single-objective search heuristics. The mutation-based NSGA-II with population size N (sufficiently large) computes the Pareto front of this problem in time O(N n k ), where n is the length of the bit-string representation and k is the jump size, a difficulty parameter of the problem [DQ22]. The authors say that they believe this bound to be tight, but prove a lower bound of Ω(n k ) only.…”

Section: Introductionmentioning

confidence: 99%

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Runtime Analysis for the NSGA-II: Provable Speed-Ups From Crossover

Doerr¹,

Qu²

2022

Preprint

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Very recently, the first mathematical runtime analyses for the NSGA-II, the most common multi-objective evolutionary algorithm, have been conducted (Zheng, Liu, Doerr (AAAI 2022)). Continuing this research direction, we prove that the NSGA-II optimizes the One-JumpZeroJump benchmark asymptotically faster when crossover is employed. This is the first time such an advantage of crossover is proven for the NSGA-II. Our arguments can be transferred to singleobjective optimization. They then prove that crossover can speedup the (µ + 1) genetic algorithm in a different way and more pronounced than known before. Our experiments confirm the added value of crossover and show that the observed speed-ups are even larger than what our proofs can guarantee.

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Section: Previous Workmentioning

confidence: 99%

Section: Benchmark Problemsmentioning

confidence: 99%

Section: Lemma 3 Consider An Iteration T Of the Nsga-ii Algorithm Opt...mentioning

confidence: 99%