Evolutionary multimodal optimization: Difference between revisions

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{{Evolutionary algorithms}}

In [[applied mathematics]], '''multimodal optimization''' deals with [[Mathematical optimization|optimization]] tasks that involve finding all or most of the multiple (at least locally optimal) solutions of a problem, as opposed to a single best solution. Evolutionary multimodal optimization is a branch of [[evolutionary computation]], which is closely related to [[machine learning]]. Wong provides a short survey,<ref>Wong, K. C. (2015), [https://arxiv.org/abs/1508.00457 Evolutionary Multimodal Optimization: A Short Survey] arXiv preprint arXiv:1508.00457</ref> wherein the chapter of Shir<ref>Shir, O.M. (2012), [https://link.springer.com/book/10.1007/978-3-540-92910-9 Niching in Evolutionary Algorithms] {{Webarchive|url=https://web.archive.org/web/20160304110426/http://link.springer.com/book/10.1007/978-3-540-92910-9 |date=2016-03-04 }}</ref> and the book of Preuss<ref>Preuss, Mike (2015), [https://www.springer.com/de/book/9783319074061 Multimodal Optimization by Means of Evolutionary Algorithms]</ref> cover the topic in more detail.

Evolutionary multimodal optimization is a branch of [[evolutionary computation]], which is closely related to [[machine learning]]. Wong provides a short survey,<ref>Wong, K. C. (2015), [https://arxiv.org/abs/1508.00457 Evolutionary Multimodal Optimization: A Short Survey] arXiv preprint arXiv:1508.00457</ref> wherein the chapter of Shir<ref>Shir, O.M. (2012), [https://link.springer.com/book/10.1007/978-3-540-92910-9 Niching in Evolutionary Algorithms] {{Webarchive|url=https://web.archive.org/web/20160304110426/http://link.springer.com/book/10.1007/978-3-540-92910-9 |date=2016-03-04 }}</ref> and the book of Preuss<ref>Preuss, Mike (2015), [https://www.springer.com/de/book/9783319074061 Multimodal Optimization by Means of Evolutionary Algorithms]</ref> cover the topic in more detail.

== Motivation ==

Knowledge of multiple solutions to an optimization task is especially helpful in engineering, when due to physical (and/or cost) constraints, the best results may not always be realizable. In such a scenario, if multiple solutions (locally and/or globally optimal) are known, the implementation can be quickly switched to another solution and still obtain the best possible system performance. Multiple solutions could also be analyzed to discover hidden properties (or relationships) of the underlying optimization problem, which makes them important for obtaining [[___domain knowledge]]. In addition, the algorithms for multimodal optimization usually not only locate multiple optima in a single run, but also preserve their population diversity, resulting in their global optimization ability on multimodal functions. Moreover, the techniques for multimodal optimization are usually borrowed as diversity maintenance techniques to other problems.<ref>Wong, K. C. et al. (2012), [https://dx.doi.org/10.1016/j.ins.2011.12.016 Evolutionary multimodal optimization using the principle of locality] Information Sciences</ref><ref>{{Cite journal |last1=Jiang |first1=Yi |last2=Zhan |first2=Zhi-Hui |last3=Tan |first3=Kay Chen |last4=Zhang |first4=Jun |date=April 2023 |title=Optimizing Niche Center for Multimodal Optimization Problems |journal=IEEE Transactions on Cybernetics |volume=53 |issue=4 |pages=2544–2557 |doi=10.1109/TCYB.2021.3125362 |issn=2168-2267|doi-access=free |pmid=34919526 }}</ref>

In addition, the algorithms for multimodal optimization usually not only locate multiple optima in a single run, but also preserve their population diversity, resulting in their global optimization ability on multimodal functions. Moreover, the techniques for multimodal optimization are usually borrowed as diversity maintenance techniques to other problems.<ref>Wong, K. C. et al. (2012), [https://dx.doi.org/10.1016/j.ins.2011.12.016 Evolutionary multimodal optimization using the principle of locality] Information Sciences</ref>

== Background ==

Classical techniques of optimization would need multiple restart points and multiple runs in the hope that a different solution may be discovered every run, with no guarantee however. [[Evolutionary algorithm]]s (EAs) due to their population based approach, provide a natural advantage over classical optimization techniques. They maintain a population of possible solutions, which are processed every generation, and if the multiple solutions can be preserved over all these generations, then at termination of the algorithm we will have multiple good solutions, rather than only the best solution. Note that this is against the natural tendency of classical optimization techniques, which will always converge to the best solution, or a sub-optimal solution (in a rugged, “badly behaving” function).''' Finding''' and '''maintenance''' of multiple solutions is wherein lies the challenge of using EAs for multi-modal optimization. '''Niching'''<ref>Mahfoud, S. W. (1995), "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.8270&rep=rep1&type=pdf Niching methods for genetic algorithms]"</ref> is a generic term referred to as the technique of finding and preserving multiple stable ''niches'', or favorable parts of the solution space possibly around multiple solutions, so as to prevent convergence to a single solution.

The field of [[Evolutionary algorithm]]s encompasses [[genetic algorithm]]s (GAs), [[evolution strategy]] (ES), [[differential evolution]] (DE), [[particle swarm optimization]] (PSO), and other methods. Attempts have been made to solve multi-modal optimization in all these realms and most, if not all the various methods implement niching in some form or the other.

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The application of multimodal optimization within ES was not explicit for many years, and has been explored only recently.

A niching framework utilizing derandomized ES was introduced by Shir,<ref>Shir, O.M. (2008), "[https://openaccess.leidenuniv.nl/handle/1887/12981 Niching in Derandomized Evolution Strategies and its Applications in Quantum Control]"</ref> '''proposing the [[CMA-ES]] as a niching optimizer for the first time'''. The underpinning of that framework was the selection of a peak individual per subpopulation in each generation, followed by its sampling to produce the consecutive dispersion of search-points. The ''biological analogy'' of this machinery is an ''alpha-male'' winning all the imposed competitions and dominating thereafter its ''ecological niche'', which then obtains all the sexual resources therein to generate its offspring.

Recently, an evolutionary [[multiobjective optimization]] (EMO) approach was proposed,<ref>Deb, K., Saha, A. (2010) "[https://dl.acm.org/doi/pdf/10.1145/1830483.1830568 Finding Multiple Solutions for Multimodal Optimization Problems Using a Multi-Objective Evolutionary Approach]" (GECCO 2010, In press)</ref> in which a suitable second objective is added to the originally single objective multimodal optimization problem, so that the multiple solutions form a '' weak pareto-optimal'' front. Hence, the multimodal optimization problem can be solved for its multiple solutions using an EMO algorithm. Improving upon their work,<ref>Saha, A., Deb, K. (2010) "A Bi-criterion Approach to Multimodal Optimization: Self-adaptive Approach " (Lecture Notes in Computer Science, 2010, Volume 6457/2010, 95–104)</ref> the same authors have made their algorithm self-adaptive, thus eliminating the need for pre-specifying the parameters.

An approach that does not use any radius for separating the population into subpopulations (or species) but employs the space topology instead is proposed in.<ref>C. Stoean, M. Preuss, R. Stoean, D. Dumitrescu (2010) [httphttps://ieeexplore.ieee.org/xpldocument/freeabs_all.jsp5491155/;jsessionid=F5D1F368440FA8710935CB745020610C?arnumber=5491155 Multimodal Optimization by means of a Topological Species Conservation Algorithm]. In IEEE Transactions on Evolutionary Computation, Vol. 14, Issue 6, pages 842–864, 2010.</ref>

[[File:GA-Multi-modal.ogv|thumbtime=1|thumb |350px |alt= Finding multiple optima using Genetic Algorithms in a Multi-modal optimization task| Finding multiple optima using genetic algorithms in a multi-modal optimization task. (The algorithm demonstrated in this demo is the one proposed by Deb, Saha in the multi-objective approach to multimodal optimization.)]]

== Multimodal optimization using DE ==

The niching methods used in GAs have also been explored with success in the DE community. DE based local selection and global selection approaches have also been attempted for solving multi-modal problems. DE's coupled with local search algorithms (Memetic DE) have been explored as an approach to solve multi-modal problems.

For a comprehensive treatment of multimodal optimization methods in DE, refer the Ph.D thesis Ronkkonen, J. (2009). ''Continuous Multimodal Global Optimization with Differential Evolution Based Methods''.<ref>Ronkkonen,J., (2009). [https://oa.doria.fi/bitstream/handle/10024/50498/isbn%209789522148520.pdf Continuous Multimodal Global Optimization with Differential Evolution Based Methods] {{Webarchive|url=https://web.archive.org/web/20141225150016/https://oa.doria.fi/bitstream/handle/10024/50498/isbn%209789522148520.pdf |date=2014-12-25 }}</ref>

== References ==

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{{refbegin}}

* D. Goldberg and J. Richardson. (1987) "[https://books.google.com/books?hl=en&lr=&id=MYJ_AAAAQBAJ&oi=fnd&pg=PA41&dq=%22Genetic+algorithms+with+sharing+for+multimodal+function+optimization%22&otspg=XwsKxp3zHA&sig=5xS0qPl-83-h3FlY2vGoPN0kmK8#v=onepage&q=%22Genetic%20algorithms%20with%20sharing%20for%20multimodal%20function%20optimization%22&f=falsePA41 Genetic algorithms with sharing for multimodal function optimization]". In Proceedings of the Second International Conference on Genetic Algorithms on Genetic algorithms and their application table of contents, pages 41–49. L. Erlbaum Associates Inc. Hillsdale, NJ, USA, 1987.

* A. Petrowski. (1996) "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.33.8027&rep=rep1&type=pdf A clearing procedure as a niching method for genetic algorithms]". In Proceedings of the 1996 IEEE International Conference on Evolutionary Computation, pages 798–803. Citeseer, 1996.

* Deb, K., (2001) "Multi-objective Optimization using Evolutionary Algorithms", Wiley ([https://books.google.com/books?id=OSTn4GSy2uQC&printsec=frontcover&dqq=multi+objective+optimization&source=bl&ots=tCmpqyNlj0&sig=r00IYlDnjaRVU94DvotX-I5mVCI&hl=en&ei=fHnNS4K5IMuLkAWJ8OgS&sa=X&oi=book_result&ct=result&resnum=8&ved=0CD0Q6AEwBw#v=onepage&q&f=false Google Books)]

* F. Streichert, G. Stein, H. Ulmer, and A. Zell. (2004) "[http://neuro.bstu.by/ai/To-dom/My_research/Papers-0/For-courses/Niche/streichert03clustering.pdf A clustering based niching EA for multimodal search spaces]". Lecture Notes in Computer Science, pages 293–304, 2004.

* Singh, G., Deb, K., (2006) "[http://repository.ias.ac.in/81664/1/94-p.pdf Comparison of multi-modal optimization algorithms based on evolutionary algorithms]". In Proceedings of the 8th annual conference on Genetic and evolutionary computation, pages 8–12. ACM, 2006.

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* Saha, A., Deb, K. (2010), [https://doi.org/10.1007%2F978-3-642-17298-4_10 A Bi-criterion Approach to Multimodal Optimization: Self-adaptive Approach. SEAL 2010: 95–104]

* Shir, O.M., Emmerich, M., Bäck, T. (2010), [http://www.mitpressjournals.org/doi/abs/10.1162/evco.2010.18.1.18104#.VoDu4l6Y7ro Adaptive Niche Radii and Niche Shapes Approaches for Niching with the CMA-ES. Evolutionary Computation Vol. 18, No. 1, pp.&nbsp; 97-126.]

* C. Stoean, M. Preuss, R. Stoean, D. Dumitrescu (2010) [httphttps://ieeexplore.ieee.org/xpldocument/freeabs_all.jsp5491155/;jsessionid=B49A922A84DC705DEF017B4E093B0894?arnumber=5491155 Multimodal Optimization by means of a Topological Species Conservation Algorithm]. In IEEE Transactions on Evolutionary Computation, Vol. 14, Issue 6, pages 842–864, 2010.

* S. Das, S. Maity, B-Y Qu, P. N. Suganthan, "[https://www.sciencedirect.com/science/article/pii/S221065021100023X Real-parameter evolutionary multimodal optimization — A survey of the state-of-the-art]", Vol. 1, No. 2, pp.&nbsp;71–88, Swarm and Evolutionary Computation, June 2011.

{{refend}}

== External links ==

* [https://web.archive.org/web/20100622065416/http://tracer.uc3m.es/tws/pso/multimodal.html Multi-modal optimization using Particle Swarm Optimization (PSO)]

* [https://web.archive.org/web/20160106231845/http://cs.telhai.ac.il/~ofersh/NichingES/index.htm Niching in Evolution Strategies (ES)]

* [http://ls11-www.cs.uni-dortmund.de/rudolph/multimodal/start Multimodal optimization page at Chair 11, Computer Science, TU Dortmund University]

* [http://www.epitropakis.co.uk/ieee-mmo/ IEEE CIS Task Force on Multi-modal Optimization]