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<title>Evolutionary multimodal optimization</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Evolutionary multimodal optimization</span></span>

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</style><table class="sidebar nomobile nowraplinks vcard hlist"><tbody><tr><td class="sidebar-pretitle">Part of a series on the</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Chromosome_(evolutionary_algorithm)" title="Chromosome (evolutionary algorithm)">Chromosome</a></li>

<li><a href="Fitness_function" title="Fitness function">Fitness function</a></li>

<li><a href="Genetic_operator" title="Genetic operator">Genetic operator</a>

<ul><li><a href="Crossover_(evolutionary_algorithm)" title="Crossover (evolutionary algorithm)">Crossover</a></li>

<li><a href="Mutation_(evolutionary_algorithm)" title="Mutation (evolutionary algorithm)">Mutation</a></li>

<li><a href="Selection_(evolutionary_algorithm)" title="Selection (evolutionary algorithm)">Selection</a></li></ul></li>

<li><a href="Population_model_(evolutionary_algorithm)" title="Population model (evolutionary algorithm)">Population model</a></li></ul></td>

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<a href="Genetic_algorithm" title="Genetic algorithm">Genetic algorithm</a> (GA)</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Chromosome_(genetic_algorithm)" class="mw-redirect" title="Chromosome (genetic algorithm)">Chromosome</a></li>

<li><a href="Clonal_selection_algorithm" title="Clonal selection algorithm">Clonal selection algorithm</a></li>

<li><a href="Fly_algorithm" title="Fly algorithm">Fly algorithm</a></li>

<li><a href="Genetic_fuzzy_systems" title="Genetic fuzzy systems">Genetic fuzzy systems</a></li>

<li><a href="Genetic_memory_(computer_science)" title="Genetic memory (computer science)">Genetic memory</a></li>

<li><a href="Schema_(genetic_algorithms)" title="Schema (genetic algorithms)">Schema</a></li>

<li><a href="Promoter_based_genetic_algorithm" title="Promoter based genetic algorithm">Promoter based GA</a></li></ul></td>

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<a href="Genetic_programming" title="Genetic programming">Genetic programming</a> (GP)</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Cartesian_genetic_programming" title="Cartesian genetic programming">Cartesian GP</a></li>

<li><a href="Linear_genetic_programming" title="Linear genetic programming">Linear GP</a></li>

<li><a href="Gene_expression_programming" title="Gene expression programming">Gene expression programming</a></li>

<li><a href="Grammatical_evolution" title="Grammatical evolution">Grammatical evolution</a></li>

<li><a href="Multi_expression_programming" title="Multi expression programming">Multi expression programming</a></li></ul></td>

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<a href="Differential_evolution" title="Differential evolution">Differential evolution</a></th></tr><tr><th class="sidebar-heading">

<a href="Evolution_strategy" title="Evolution strategy">Evolution strategy</a></th></tr><tr><th class="sidebar-heading">

<a href="Evolutionary_programming" title="Evolutionary programming">Evolutionary programming</a></th></tr><tr><th class="sidebar-heading">

Related topics</th></tr><tr><td class="sidebar-content hlist">

<ul><li><a href="Cellular_evolutionary_algorithm" title="Cellular evolutionary algorithm">Cellular EA</a></li>

<li><a href="Cultural_algorithm" title="Cultural algorithm">Cultural algorithm</a></li>

<li><a href="Effective_fitness" title="Effective fitness">Effective fitness</a></li>

<li><a href="Evolutionary_computation" title="Evolutionary computation">Evolutionary computation</a></li>

<li><a href="Gaussian_adaptation" title="Gaussian adaptation">Gaussian adaptation</a></li>

<li><a href="Grammar_induction#Grammatical_inference_by_genetic_algorithms" title="Grammar induction">Grammar induction</a></li>

<li><a href="Memetic_algorithm" title="Memetic algorithm">Memetic algorithm</a></li>

<li><a href="Neuroevolution" title="Neuroevolution">Neuroevolution</a></li></ul></td>

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<p>In <a href="Applied_mathematics" title="Applied mathematics">applied mathematics</a>, <b>multimodal optimization</b> deals with <a href="Mathematical_optimization" title="Mathematical optimization">optimization</a> 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 <a href="Evolutionary_computation" title="Evolutionary computation">evolutionary computation</a>, which is closely related to <a href="Machine_learning" title="Machine learning">machine learning</a>. Wong provides a short survey,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> wherein the chapter of Shir<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and the book of Preuss<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> cover the topic in more detail.

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<div class="mw-heading mw-heading2"><h2 id="Motivation">Motivation</h2></div>

<p>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 <a href="Domain_knowledge" title="Domain knowledge">domain knowledge</a>. 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.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Background">Background</h2></div>

<p>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. <a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithms</a> (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<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> is a generic term referred to as the technique of finding and preserving multiple stable <i>niches</i>, or favorable parts of the solution space possibly around multiple solutions, so as to prevent convergence to a single solution.

</p><p>The field of <a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithms</a> encompasses <a href="Genetic_algorithm" title="Genetic algorithm">genetic algorithms</a> (GAs), <a href="Evolution_strategy" title="Evolution strategy">evolution strategy</a> (ES), <a href="Differential_evolution" title="Differential evolution">differential evolution</a> (DE), <a href="Particle_swarm_optimization" title="Particle swarm optimization">particle swarm optimization</a> (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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<div class="mw-heading mw-heading2"><h2 id="Multimodal_optimization_using_genetic_algorithms/evolution_strategies">Multimodal optimization using genetic algorithms/evolution strategies</h2></div>

<p>De Jong's crowding method, Goldberg's sharing function approach, Petrowski's clearing method, restricted mating, maintaining multiple subpopulations are some of the popular approaches that have been proposed by the community. The first two methods are especially well studied, however, they do not perform explicit separation into solutions belonging to different basins of attraction.

</p><p>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,<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> proposing the <a href="CMA-ES" title="CMA-ES">CMA-ES</a> 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 <i>biological analogy</i> of this machinery is an <i>alpha-male</i> winning all the imposed competitions and dominating thereafter its <i>ecological niche</i>, which then obtains all the sexual resources therein to generate its offspring.

</p><p>Recently, an evolutionary <a href="Multiobjective_optimization" class="mw-redirect" title="Multiobjective optimization">multiobjective optimization</a> (EMO) approach was proposed,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> in which a suitable second objective is added to the originally single objective multimodal optimization problem, so that the multiple solutions form a <i> weak pareto-optimal</i> front. Hence, the multimodal optimization problem can be solved for its multiple solutions using an EMO algorithm. Improving upon their work,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> the same authors have made their algorithm self-adaptive, thus eliminating the need for pre-specifying the parameters.

</p><p>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.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Wong, K. C. (2015), <a rel="nofollow" class="external text" href="https://arxiv.org/abs/1508.00457">Evolutionary Multimodal Optimization: A Short Survey</a> arXiv preprint arXiv:1508.00457</span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Shir, O.M. (2012), <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-540-92910-9">Niching in Evolutionary Algorithms</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304110426/http://link.springer.com/book/10.1007/978-3-540-92910-9">Archived</a> 2016-03-04 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Preuss, Mike (2015), <a rel="nofollow" class="external text" href="https://www.springer.com/de/book/9783319074061">Multimodal Optimization by Means of Evolutionary Algorithms</a></span>

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<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Wong, K. C. et al. (2012), <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1016/j.ins.2011.12.016">Evolutionary multimodal optimization using the principle of locality</a> Information Sciences</span>

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</style><cite id="CITEREFJiangZhanTanZhang2023" class="citation journal cs1">Jiang, Yi; Zhan, Zhi-Hui; Tan, Kay Chen; Zhang, Jun (April 2023). <a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCYB.2021.3125362">"Optimizing Niche Center for Multimodal Optimization Problems"</a>. <i>IEEE Transactions on Cybernetics</i>. <b>53</b> (4): <span class="nowrap">2544–</span>2557. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTCYB.2021.3125362">10.1109/TCYB.2021.3125362</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2168-2267">2168-2267</a>. <a href="PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/34919526">34919526</a>.</cite></span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Mahfoud, S. W. (1995), "<a rel="nofollow" class="external text" href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.8270&rep=rep1&type=pdf">Niching methods for genetic algorithms</a>"</span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Shir, O.M. (2008), "<a rel="nofollow" class="external text" href="https://openaccess.leidenuniv.nl/handle/1887/12981">Niching in Derandomized Evolution Strategies and its Applications in Quantum Control</a>"</span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Deb, K., Saha, A. (2010) "<a rel="nofollow" class="external text" href="https://dl.acm.org/doi/pdf/10.1145/1830483.1830568">Finding Multiple Solutions for Multimodal Optimization Problems Using a Multi-Objective Evolutionary Approach</a>" (GECCO 2010, In press)</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">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)</span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">C. Stoean, M. Preuss, R. Stoean, D. Dumitrescu (2010) <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/5491155/;jsessionid=F5D1F368440FA8710935CB745020610C?arnumber=5491155">Multimodal Optimization by means of a Topological Species Conservation Algorithm</a>. In IEEE Transactions on Evolutionary Computation, Vol. 14, Issue 6, pages 842–864, 2010.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

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<ul><li>D. Goldberg and J. Richardson. (1987) "<a rel="nofollow" class="external text" href="https://books.google.com/books?id=MYJ_AAAAQBAJ&dq=%22Genetic+algorithms+with+sharing+for+multimodal+function+optimization%22&pg=PA41">Genetic algorithms with sharing for multimodal function optimization</a>". 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.</li>

<li>A. Petrowski. (1996) "<a rel="nofollow" class="external text" href="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</a>". In Proceedings of the 1996 IEEE International Conference on Evolutionary Computation, pages 798–803. Citeseer, 1996.</li>

<li>Deb, K., (2001) "Multi-objective Optimization using Evolutionary Algorithms", Wiley (<a rel="nofollow" class="external text" href="https://books.google.com/books?id=OSTn4GSy2uQC&q=multi+objective+optimization">Google Books)</a></li>

<li>F. Streichert, G. Stein, H. Ulmer, and A. Zell. (2004) "<a rel="nofollow" class="external text" href="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</a>". Lecture Notes in Computer Science, pages 293–304, 2004.</li>

<li>Singh, G., Deb, K., (2006) "<a rel="nofollow" class="external text" href="http://repository.ias.ac.in/81664/1/94-p.pdf">Comparison of multi-modal optimization algorithms based on evolutionary algorithms</a>". In Proceedings of the 8th annual conference on Genetic and evolutionary computation, pages 8–12. ACM, 2006.</li>

<li>Ronkkonen, J., (2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20141225150016/https://oa.doria.fi/bitstream/handle/10024/50498/isbn%209789522148520.pdf">Continuous Multimodal Global Optimization with Differential Evolution Based Methods</a></li>

<li>Wong, K. C., (2009). <a rel="nofollow" class="external text" href="http://portal.acm.org/citation.cfm?id=1570027">An evolutionary algorithm with species-specific explosion for multimodal optimization. GECCO 2009: 923–930</a></li>

<li>J. Barrera and C. A. C. Coello. "<a rel="nofollow" class="external text" href="http://delta.cs.cinvestav.mx/~ccoello/EMOO/barrera09a.pdf.gz">A Review of Particle Swarm Optimization Methods used for Multimodal Optimization</a>", pages 9–37. Springer, Berlin, November 2009.</li>

<li>Wong, K. C., (2010). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-12239-2_50">Effect of Spatial Locality on an Evolutionary Algorithm for Multimodal Optimization. EvoApplications (1) 2010: 481–490</a></li>

<li>Deb, K., Saha, A. (2010) <a rel="nofollow" class="external text" href="http://portal.acm.org/citation.cfm?id=1830483.1830568">Finding Multiple Solutions for Multimodal Optimization Problems Using a Multi-Objective Evolutionary Approach. GECCO 2010: 447–454</a></li>

<li>Wong, K. C., (2010). <a rel="nofollow" class="external text" href="http://portal.acm.org/citation.cfm?id=1830483.1830513">Protein structure prediction on a lattice model via multimodal optimization techniques. GECCO 2010: 155–162</a></li>

<li>Saha, A., Deb, K. (2010), <a rel="nofollow" class="external text" href="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</a></li>

<li>Shir, O.M., Emmerich, M., Bäck, T. (2010), <a rel="nofollow" class="external text" href="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.  97-126.</a></li>

<li>C. Stoean, M. Preuss, R. Stoean, D. Dumitrescu (2010) <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/5491155/;jsessionid=B49A922A84DC705DEF017B4E093B0894?arnumber=5491155">Multimodal Optimization by means of a Topological Species Conservation Algorithm</a>. In IEEE Transactions on Evolutionary Computation, Vol. 14, Issue 6, pages 842–864, 2010.</li>

<li>S. Das, S. Maity, B-Y Qu, P. N. Suganthan, "<a rel="nofollow" class="external text" href="https://www.sciencedirect.com/science/article/pii/S221065021100023X">Real-parameter evolutionary multimodal optimization — A survey of the state-of-the-art</a>", Vol. 1, No. 2, pp. 71–88, Swarm and Evolutionary Computation, June 2011.</li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100622065416/http://tracer.uc3m.es/tws/pso/multimodal.html">Multi-modal optimization using Particle Swarm Optimization (PSO)</a></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20160106231845/http://cs.telhai.ac.il/~ofersh/NichingES/index.htm">Niching in Evolution Strategies (ES)</a></li>

<li><a rel="nofollow" class="external text" href="http://ls11-www.cs.uni-dortmund.de/rudolph/multimodal/start">Multimodal optimization page at Chair 11, Computer Science, TU Dortmund University</a></li>

<li><a rel="nofollow" class="external text" href="http://www.epitropakis.co.uk/ieee-mmo/">IEEE CIS Task Force on Multi-modal Optimization</a></li></ul>

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</style></div><div role="navigation" class="navbox" aria-labelledby="Optimization:_Algorithms,_methods,_and_heuristics381" style="padding:3px"><table class="nowraplinks hlist mw-collapsible expanded navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="3"><div id="Optimization:_Algorithms,_methods,_and_heuristics381" style="font-size:114%;margin:0 4em"><a href="Mathematical_optimization" title="Mathematical optimization">Optimization</a>: <a href="Optimization_algorithm" class="mw-redirect" title="Optimization algorithm">Algorithms</a>, <a href="Iterative_method" title="Iterative method">methods</a>, and <a href="Heuristic_algorithm" class="mw-redirect" title="Heuristic algorithm">heuristics</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Unconstrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="Nonlinear_programming" title="Nonlinear programming">Unconstrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Function_(mathematics)" title="Function (mathematics)">Functions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Golden-section_search" title="Golden-section search">Golden-section search</a></li>

<li><a href="Powell's_method" title="Powell's method">Powell's method</a></li>

<li><a href="Line_search" title="Line search">Line search</a></li>

<li><a href="Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead method</a></li>

<li><a href="Successive_parabolic_interpolation" title="Successive parabolic interpolation">Successive parabolic interpolation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Gradient" title="Gradient">Gradients</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Local_convergence" title="Local convergence">Convergence</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Trust_region" title="Trust region">Trust region</a></li>

<li><a href="Wolfe_conditions" title="Wolfe conditions">Wolfe conditions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Quasi-Newton_method" title="Quasi-Newton method">Quasi–Newton</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Berndt%E2%80%93Hall%E2%80%93Hall%E2%80%93Hausman_algorithm" title="Berndt–Hall–Hall–Hausman algorithm">Berndt–Hall–Hall–Hausman</a></li>

<li><a href="Broyden%E2%80%93Fletcher%E2%80%93Goldfarb%E2%80%93Shanno_algorithm" title="Broyden–Fletcher–Goldfarb–Shanno algorithm">Broyden–Fletcher–Goldfarb–Shanno</a> and <a href="Limited-memory_BFGS" title="Limited-memory BFGS">L-BFGS</a></li>

<li><a href="Davidon%E2%80%93Fletcher%E2%80%93Powell_formula" title="Davidon–Fletcher–Powell formula">Davidon–Fletcher–Powell</a></li>

<li><a href="Symmetric_rank-one" title="Symmetric rank-one">Symmetric rank-one (SR1)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Iterative_method" title="Iterative method">Other methods</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Nonlinear_conjugate_gradient_method" title="Nonlinear conjugate gradient method">Conjugate gradient</a></li>

<li><a href="Gauss%E2%80%93Newton_algorithm" title="Gauss–Newton algorithm">Gauss–Newton</a></li>

<li><a href="Gradient_descent" title="Gradient descent">Gradient</a></li>

<li><a href="Mirror_descent" title="Mirror descent">Mirror</a></li>

<li><a href="Levenberg%E2%80%93Marquardt_algorithm" title="Levenberg–Marquardt algorithm">Levenberg–Marquardt</a></li>

<li><a href="Powell's_dog_leg_method" title="Powell's dog leg method">Powell's dog leg method</a></li>

<li><a href="Truncated_Newton_method" title="Truncated Newton method">Truncated Newton</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Hessian_matrix" title="Hessian matrix">Hessians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Newton's_method_in_optimization" title="Newton's method in optimization">Newton's method</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Constrained_nonlinear381" style="font-size:114%;margin:0 4em"><a href="Nonlinear_programming" title="Nonlinear programming">Constrained nonlinear</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Barrier_function" title="Barrier function">Barrier methods</a></li>

<li><a href="Penalty_method" title="Penalty method">Penalty methods</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Differentiable</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Augmented_Lagrangian_method" title="Augmented Lagrangian method">Augmented Lagrangian methods</a></li>

<li><a href="Sequential_quadratic_programming" title="Sequential quadratic programming">Sequential quadratic programming</a></li>

<li><a href="Successive_linear_programming" title="Successive linear programming">Successive linear programming</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Convex_optimization381" style="font-size:114%;margin:0 4em"><a href="Convex_optimization" title="Convex optimization">Convex optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Convex_minimization" class="mw-redirect" title="Convex minimization">Convex<br> minimization</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cutting-plane_method" title="Cutting-plane method">Cutting-plane method</a></li>

<li><a href="Frank%E2%80%93Wolfe_algorithm" title="Frank–Wolfe algorithm">Reduced gradient (Frank–Wolfe)</a></li>

<li><a href="Subgradient_method" title="Subgradient method">Subgradient method</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Linear_programming" title="Linear programming">Linear</a> and<br><a href="Quadratic_programming" title="Quadratic programming">quadratic</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Linear_programming#Interior_point" title="Linear programming">Interior point</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_scaling" title="Affine scaling">Affine scaling</a></li>

<li><a href="Ellipsoid_method" title="Ellipsoid method">Ellipsoid algorithm of Khachiyan</a></li>

<li><a href="Karmarkar's_algorithm" title="Karmarkar's algorithm">Projective algorithm of Karmarkar</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Matroid" title="Matroid">Basis-</a><a href="Exchange_algorithm" class="mw-redirect" title="Exchange algorithm">exchange</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Simplex_algorithm" title="Simplex algorithm">Simplex algorithm of Dantzig</a></li>

<li><a href="Revised_simplex_method" title="Revised simplex method">Revised simplex algorithm</a></li>

<li><a href="Criss-cross_algorithm" title="Criss-cross algorithm">Criss-cross algorithm</a></li>

<li><a href="Lemke's_algorithm" title="Lemke's algorithm">Principal pivoting algorithm of Lemke</a></li>

<li><a href="Active-set_method" title="Active-set method">Active-set method</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial381" style="font-size:114%;margin:0 4em"><a href="Combinatorial_optimization" title="Combinatorial optimization">Combinatorial</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Paradigms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Approximation_algorithm" title="Approximation algorithm">Approximation algorithm</a></li>

<li><a href="Dynamic_programming" title="Dynamic programming">Dynamic programming</a></li>

<li><a href="Greedy_algorithm" title="Greedy algorithm">Greedy algorithm</a></li>

<li><a href="Integer_programming" title="Integer programming">Integer programming</a>

<ul><li><a href="Branch_and_bound" title="Branch and bound">Branch and bound</a>/<a href="Branch_and_cut" title="Branch and cut">cut</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Graph_algorithm" class="mw-redirect" title="Graph algorithm">Graph<br> algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Minimum_spanning_tree52" scope="row" class="navbox-group" style="width:1%"><a href="Minimum_spanning_tree" title="Minimum spanning tree">Minimum<br> spanning tree</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bor%C5%AFvka's_algorithm" title="Borůvka's algorithm">Borůvka</a></li>

<li><a href="Kruskal's_algorithm" title="Kruskal's algorithm">Kruskal</a></li></ul>

</div></td></tr></tbody></table><div>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th id="Shortest_path39" scope="row" class="navbox-group" style="width:1%"><a href="Shortest_path_problem" title="Shortest path problem">Shortest path</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bellman%E2%80%93Ford_algorithm" title="Bellman–Ford algorithm">Bellman–Ford</a>

<li><a href="Dijkstra's_algorithm" title="Dijkstra's algorithm">Dijkstra</a></li>

<li><a href="Floyd%E2%80%93Warshall_algorithm" title="Floyd–Warshall algorithm">Floyd–Warshall</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Flow_network" title="Flow network">Network flows</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Dinic's_algorithm" title="Dinic's algorithm">Dinic</a></li>

<li><a href="Edmonds%E2%80%93Karp_algorithm" title="Edmonds–Karp algorithm">Edmonds–Karp</a></li>

<li><a href="Ford%E2%80%93Fulkerson_algorithm" title="Ford–Fulkerson algorithm">Ford–Fulkerson</a></li>

<li><a href="Push%E2%80%93relabel_maximum_flow_algorithm" title="Push–relabel maximum flow algorithm">Push–relabel maximum flow</a></li></ul>

<ul><li><a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></li>

<li><a href="Hill_climbing" title="Hill climbing">Hill climbing</a></li>

<li><a href="Local_search_(optimization)" title="Local search (optimization)">Local search</a></li>

<li><a href="Parallel_metaheuristic" title="Parallel metaheuristic">Parallel metaheuristics</a></li>

<li><a href="Simulated_annealing" title="Simulated annealing">Simulated annealing</a></li>

<li><a href="Spiral_optimization_algorithm" title="Spiral optimization algorithm">Spiral optimization algorithm</a></li>

<li><a href="Tabu_search" title="Tabu search">Tabu search</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li><a href="Comparison_of_optimization_software" title="Comparison of optimization software">Software</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Evolutionary_computation153" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Evolutionary_computation153" style="font-size:114%;margin:0 4em"><a href="Evolutionary_computation" title="Evolutionary computation">Evolutionary computation</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main Topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Evolutionary_algorithm" title="Evolutionary algorithm">Evolutionary algorithm</a></li>

<li><a href="Evolutionary_data_mining" title="Evolutionary data mining">Evolutionary data mining</a></li>

<li><a href="Human-based_evolutionary_computation" title="Human-based evolutionary computation">Human-based evolutionary computation</a></li>

<li><a href="Interactive_evolutionary_computation" title="Interactive evolutionary computation">Interactive evolutionary computation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Algorithm" title="Algorithm">Algorithms</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cellular_evolutionary_algorithm" title="Cellular evolutionary algorithm">Cellular evolutionary algorithm</a></li>

<li><a href="CMA-ES" title="CMA-ES">Covariance Matrix Adaptation Evolution Strategy (CMA-ES)</a></li>

<li><a href="Cultural_algorithm" title="Cultural algorithm">Cultural algorithm</a></li>

<li><a href="Differential_evolution" title="Differential evolution">Differential evolution</a></li>

<li><a href="Evolutionary_programming" title="Evolutionary programming">Evolutionary programming</a></li>

<li><a href="Genetic_algorithm" title="Genetic algorithm">Genetic algorithm</a></li>

<li><a href="Genetic_programming" title="Genetic programming">Genetic programming</a></li>

<li><a href="Gene_expression_programming" title="Gene expression programming">Gene expression programming</a></li>

<li><a href="Evolution_strategy" title="Evolution strategy">Evolution strategy</a></li>

<li><a href="Natural_evolution_strategy" title="Natural evolution strategy">Natural evolution strategy</a></li>

<li><a href="Neuroevolution" title="Neuroevolution">Neuroevolution</a></li>

<li><a href="Learning_classifier_system" title="Learning classifier system">Learning classifier system</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related techniques</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Swarm_intelligence" title="Swarm intelligence">Swarm intelligence</a></li>

<li><a href="Ant_colony_optimization" class="mw-redirect" title="Ant colony optimization">Ant colony optimization</a></li>

<li><a href="Bees_algorithm" title="Bees algorithm">Bees algorithm</a></li>

<li><a href="Cuckoo_search" title="Cuckoo search">Cuckoo search</a></li>

<li><a href="Particle_swarm_optimization" title="Particle swarm optimization">Particle swarm optimization</a></li>

<li><a href="Bacterial_Colony_Optimization" class="mw-redirect" title="Bacterial Colony Optimization">Bacterial Colony Optimization</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Metaheuristic" title="Metaheuristic">Metaheuristic methods</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Firefly_algorithm" title="Firefly algorithm">Firefly algorithm</a></li>

<li><a href="Harmony_search" class="mw-redirect" title="Harmony search">Harmony search</a></li>

<li><a href="Gaussian_adaptation" title="Gaussian adaptation">Gaussian adaptation</a></li>

<li><a href="Memetic_algorithm" title="Memetic algorithm">Memetic algorithm</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Artificial_development" title="Artificial development">Artificial development</a></li>

<li><a href="Artificial_intelligence" title="Artificial intelligence">Artificial intelligence</a></li>

<li><a href="Artificial_life" title="Artificial life">Artificial life</a></li>

<li><a href="Digital_organism" title="Digital organism">Digital organism</a></li>

<li><a href="Evolutionary_robotics" title="Evolutionary robotics">Evolutionary robotics</a></li>

<li><a href="Fitness_function" title="Fitness function">Fitness function</a></li>

<li><a href="Fitness_landscape" title="Fitness landscape">Fitness landscape</a></li>

<li><a href="Fitness_approximation" title="Fitness approximation">Fitness approximation</a></li>

<li><a href="Genetic_operators" class="mw-redirect" title="Genetic operators">Genetic operators</a></li>

<li><a href="Interactive_evolutionary_computation" title="Interactive evolutionary computation">Interactive evolutionary computation</a></li>

<li><a href="No_free_lunch_in_search_and_optimization" title="No free lunch in search and optimization">No free lunch in search and optimization</a></li>

<li><a href="Machine_learning" title="Machine learning">Machine learning</a></li>

<li><a href="Mating_pool" title="Mating pool">Mating pool</a></li>

<li><a href="Premature_convergence" title="Premature convergence">Premature convergence</a></li>

<li><a href="Program_synthesis" title="Program synthesis">Program synthesis</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Academic_journal" title="Academic journal">Journals</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Evolutionary_Computation_(journal)" title="Evolutionary Computation (journal)">Evolutionary Computation (journal)</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-15" href="https://en.wikipedia.org/wiki/?title=Evolutionary_multimodal_optimization&oldid=1285689656">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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