Oliveto, P.S., Sudholt, D. and Witt, C. (2022) Tight bounds on the expected runtime of a standard steady state genetic algorithm. Algorithmica, 84 (6). pp. 1603-1658. ISSN 0178-4617
Abstract
Recent progress in the runtime analysis of evolutionary algorithms (EAs) has allowed the derivation of upper bounds on the expected runtime of standard steady-state genetic algorithms (GAs). These upper bounds have shown speed-ups of the GAs using crossover and mutation over the same algorithms that only use mutation operators (i.e., steady-state EAs) both for standard unimodal (i.e., ONEMAX) and multimodal (i.e., JUMP) benchmark functions. The bounds suggest that populations are beneficial to the GA as well as higher mutation rates than the default 1/n rate. However, making rigorous claims was not possible because matching lower bounds were not available. Proving lower bounds on crossover-based EAs is a notoriously difficult task as it is hard to capture the progress that a diverse population can make. We use a potential function approach to prove a tight lower bound on the expected runtime of the (2+1) GA for ONEMAX for all mutation rates c/n with c<1.422. This provides the last piece of the puzzle that completes the proof that larger population sizes improve the performance of the standard steady-state GA for ONEMAX for various mutation rates, and it proves that the optimal mutation rate for the (2+1) GA on ONEMAX is (97−−√−5)/(4n)≈1.2122/n.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © The Author(s) 2021. Open Access: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
Keywords: | Evolutionary algorithms; Runtime analysis; Crossover; Lower bounds |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Funding Information: | Funder Grant number ENGINEERING AND PHYSICAL SCIENCE RESEARCH COUNCIL EP/M004252/1 European Cooperation in Science and Technology CA15140 ENGINEERING AND PHYSICAL SCIENCE RESEARCH COUNCIL UNSPECIFIED |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 13 Jan 2022 16:27 |
Last Modified: | 31 May 2022 13:46 |
Status: | Published |
Publisher: | Springer |
Refereed: | Yes |
Identification Number: | 10.1007/s00453-021-00893-w |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:182523 |
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