Improved time complexity analysis of the Simple Genetic Algorithm

Many real optimisation problems lead to multimodal domains and so require the identification of multiple optima. Niching methods have been developed to maintain the population diversity, to investigate many peaks in parallel and to reduce the effect of genetic drift. Using rigorous runtime analysis, we analyse for the first time two well known niching methods: probabilistic crowding and restricted tournament selection (RTS). We incorporate both methods into a (µ+1) EA on the bimodal function Twomax where the goal is to find two optima at opposite ends of the search space. In probabilistic crowding, the offspring compete with their parents and the survivor is chosen proportionally to its fitness. On Twomax probabilistic crowding fails to find any reasonable solution quality even in exponential time. In RTS the offspring compete against the closest individual amongst w (window size) individuals. We prove that RTS fails if w is too small, leading to exponential times with high probability. However, if w is chosen large enough, it finds both optima for Twomax in time O(µn log n) with high probability. Our theoretical results are accompanied by experimental studies that match the theoretical results and also shed light on parameters not covered by the theoretical results.

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“…The Simple Genetic Algorithm (SGA) has been analysed with fitnessproportional selection for parent selection in [16,19,20].…”

Section: : End Whilementioning

confidence: 99%

Runtime analysis of probabilistic crowding and restricted tournament selection for bimodal optimisation

Osuna

Sudholt

2018

Proceedings of the Genetic and Evolutionary Computation Conference

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“…In [16], the lemma is only stated for ζ = 1. However, introducing the constant factor does not change the lemmas's proof at all.…”

Section: Lower Bound On the Variance Of The Potential Changementioning

confidence: 99%

Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax

Krejca

Witt

2017

Proceedings of the 14th ACM/SIGEVO Conference on Foundations of Genetic Algorithms

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The Univariate Marginal Distribution Algorithm (UMDA), a popular estimation of distribution algorithm, is studied from a run time perspective. On the classical OneMax benchmark function, a lower bound of Ω(µ √ n + n log n), where µ is the population size, on its expected run time is proved. This is the first direct lower bound on the run time of the UMDA. It is stronger than the bounds that follow from general black-box complexity theory and is matched by the run time of many evolutionary algorithms. The results are obtained through advanced analyses of the stochastic change of the frequencies of bit values maintained by the algorithm, including carefully designed potential functions. These techniques may prove useful in advancing the field of run time analysis for estimation of distribution algorithms in general.

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“…The generic approach of approximating binomial distributions via normal distributions is not often used in the theory of randomized search. In [OW15], the Berry-Esseen inequality was employed to prove a result similar to Lemma 10.16 for the special case of binomial distributions. Unlike many other proof relying on the normal approximation, this proof is quite short and elegant.…”

Section: Approximation Via the Normal Distributionmentioning

confidence: 99%

Probabilistic Tools for the Analysis of Randomized Optimization Heuristics

Doerr¹

2019

Natural Computing Series

120

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This chapter collects several probabilistic tools that proved to be useful in the analysis of randomized search heuristics. This includes classic material like Markov, Chebyshev and Chernoff inequalities, but also lesser known topics like stochastic domination and coupling or Chernoff bounds for geometrically distributed random variables and for negatively correlated random variables. Most of the results presented here have appeared previously, some, however, only in recent conference publications. While the focus is on collecting tools for the analysis of randomized search heuristics, many of these may be useful as well in the analysis of classic randomized algorithms or discrete random structures.

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Improved time complexity analysis of the Simple Genetic Algorithm

Cited by 121 publications

References 22 publications

Runtime analysis of probabilistic crowding and restricted tournament selection for bimodal optimisation

Runtime analysis of probabilistic crowding and restricted tournament selection for bimodal optimisation

Lower Bounds on the Run Time of the Univariate Marginal Distribution Algorithm on OneMax

Probabilistic Tools for the Analysis of Randomized Optimization Heuristics

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