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Stochastic optimization

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</style><div role="note" class="hatnote navigation-not-searchable">This article is about <a href="Iterative_method" title="Iterative method">iterative methods</a>. For the modeling (and optimization) of decisions under uncertainty, see <a href="Stochastic_programming" title="Stochastic programming">stochastic programming</a>. For the context of control theory, see <a href="Stochastic_control" title="Stochastic control">stochastic control</a>.</div>

Stochastic optimization (SO) are <a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">optimization</a> <a href="Iterative_method" title="Iterative method">methods</a> that generate and use <a href="Random_variable" title="Random variable">random variables</a>. For <a href="Stochastic" title="Stochastic">stochastic</a> optimization problems, the <a href="Objective_function" class="mw-redirect" title="Objective function">objective functions</a> or <a href="Constraint_(mathematics)" title="Constraint (mathematics)">constraints</a> are random. Stochastic optimization also include methods with random <a href="Iterative_method" title="Iterative method">iterates</a>. Some hybrid methods use random iterates to solve stochastic problems, combining both meanings of stochastic optimization.<a href="#cite_note-spall2003-1">[1]</a>

Stochastic optimization methods generalize <a href="Deterministic_system_(mathematics)" class="mw-redirect" title="Deterministic system (mathematics)">deterministic</a> methods for deterministic problems.

<div class="mw-heading mw-heading2"><h2 id="Methods_for_stochastic_functions">Methods for stochastic functions</h2></div>

Partly random input data arise in such areas as real-time estimation and control, simulation-based optimization where <a href="Monte_Carlo_method" title="Monte Carlo method">Monte Carlo simulations</a> are run as estimates of an actual system,<a href="#cite_note-fu2002-2">[2]</a><a href="#cite_note-3">[3]</a> and problems where there is experimental (random) error in the measurements of the criterion. In such cases, knowledge that the function values are contaminated by random "noise" leads naturally to algorithms that use <a href="Statistics" title="Statistics">statistical inference</a> tools to estimate the "true" values of the function and/or make statistically optimal decisions about the next steps. Methods of this class include:

<ul><li><a href="Stochastic_approximation" title="Stochastic approximation">stochastic approximation</a> (SA), by <a href="Herbert_Robbins" title="Herbert Robbins">Robbins</a> and Monro (1951)<a href="#cite_note-RM1951-4">[4]</a></li>

<li><a href="Stochastic_gradient_descent" title="Stochastic gradient descent">stochastic gradient descent</a></li>

<li><a href="Finite-difference_stochastic_approximation" class="mw-redirect" title="Finite-difference stochastic approximation">finite-difference SA</a> by <a href="Jack_Kiefer_(mathematician)" class="mw-redirect" title="Jack Kiefer (mathematician)">Kiefer</a> and <a href="Jacob_Wolfowitz" title="Jacob Wolfowitz">Wolfowitz</a> (1952)<a href="#cite_note-KW1952-5">[5]</a></li>

<li><a href="Simultaneous_perturbation_stochastic_approximation" title="Simultaneous perturbation stochastic approximation">simultaneous perturbation SA</a> by Spall (1992)<a href="#cite_note-spall1992-6">[6]</a></li>

<li><a href="Scenario_optimization" title="Scenario optimization">scenario optimization</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Randomized_search_methods">Randomized search methods</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Metaheuristic" title="Metaheuristic">Metaheuristic</a></div>

On the other hand, even when the <a href="Data_set" title="Data set">data set</a> consists of precise measurements, some methods introduce randomness into the search-process to accelerate progress.<a href="#cite_note-7">[7]</a> Such randomness can also make the method less sensitive to modeling errors. Another advantage is that randomness into the search-process can be used for obtaining interval estimates of the minimum of a function via extreme value statistics.<a href="#cite_note-deCarvalho2011-8">[8]</a><a href="#cite_note-deCarvalho2012-9">[9]</a>

Further, the injected randomness may enable the method to escape a local optimum and eventually to approach a global optimum. Indeed, this <a href="Randomized_algorithm" title="Randomized algorithm">randomization</a> principle is known to be a simple and effective way to obtain algorithms with almost certain good performance uniformly across many data sets, for many sorts of problems. Stochastic optimization methods of this kind include:

<ul><li><a href="Simulated_annealing" title="Simulated annealing">simulated annealing</a> by S. Kirkpatrick, C. D. Gelatt and M. P. Vecchi (1983)<a href="#cite_note-kirk1983-10">[10]</a></li>

<li><a href="Quantum_annealing" title="Quantum annealing">quantum annealing</a></li>

<li>Probability Collectives by D.H. Wolpert, S.R. Bieniawski and D.G. Rajnarayan (2011)<a href="#cite_note-wolp2011-11">[11]</a></li>

<li><a href="Reactive_Search_Optimization" class="mw-redirect" title="Reactive Search Optimization">reactive search optimization (RSO)</a> by <a href="Roberto_Battiti" title="Roberto Battiti">Roberto Battiti</a>, G. Tecchiolli (1994),<a href="#cite_note-BattitiTecchiolli94-12">[12]</a> recently reviewed in the reference book <a href="#cite_note-BattitiBrunatoMascia2008-13">[13]</a></li>

<li><a href="Cross-entropy_method" title="Cross-entropy method">cross-entropy method</a> by <a href="Reuven_Rubinstein" title="Reuven Rubinstein">Rubinstein</a> and <a href="Dirk_Kroese" title="Dirk Kroese">Kroese</a> (2004)<a href="#cite_note-rubkro-14">[14]</a></li>

<li><a href="Random_search" title="Random search">random search</a> by <a href="Anatoly_Zhigljavsky" title="Anatoly Zhigljavsky">Anatoly Zhigljavsky</a> (1991)<a href="#cite_note-zhigl1991-15">[15]</a></li>

<li>Informational search <a href="#cite_note-16">[16]</a></li>

<li><a href="Stochastic_tunneling" title="Stochastic tunneling">stochastic tunneling</a><a href="#cite_note-wenz1999-17">[17]</a></li>

<li><a href="Parallel_tempering" title="Parallel tempering">parallel tempering</a> a.k.a. replica exchange<a href="#cite_note-mari1992-18">[18]</a></li>

<li><a href="Stochastic_hill_climbing" title="Stochastic hill climbing">stochastic hill climbing</a></li>

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

<li><a href="Evolutionary_algorithms" class="mw-redirect" title="Evolutionary algorithms">evolutionary algorithms</a>

<ul><li><a href="Genetic_algorithms" class="mw-redirect" title="Genetic algorithms">genetic algorithms</a> by Holland (1975)<a href="#cite_note-gold1991-19">[19]</a></li>

<li><a href="Evolution_strategies" class="mw-redirect" title="Evolution strategies">evolution strategies</a></li></ul></li>

<li>cascade object optimization & modification algorithm (2016)<a href="#cite_note-20">[20]</a></li></ul>

In contrast, some authors have argued that randomization can only improve a deterministic algorithm if the deterministic algorithm was poorly designed in the first place.<a href="#cite_note-21">[21]</a>

<a href="Fred_W._Glover" title="Fred W. Glover">Fred W. Glover</a><a href="#cite_note-22">[22]</a> argues that reliance on random elements may prevent the development of more intelligent and better deterministic components. The way in which results of stochastic optimization algorithms are usually presented (e.g., presenting only the average, or even the best, out of N runs without any mention of the spread), may also result in a positive bias towards randomness.

<ul><li><a href="Global_optimization" title="Global optimization">Global optimization</a></li>

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

<li><a href="Scenario_optimization" title="Scenario optimization">Scenario optimization</a></li>

<li><a href="Gaussian_process" title="Gaussian process">Gaussian process</a></li>

<li><a href="State_Space_Model" class="mw-redirect" title="State Space Model">State Space Model</a></li>

<li><a href="Model_predictive_control" title="Model predictive control">Model predictive control</a></li>

<li><a href="Nonlinear_programming" title="Nonlinear programming">Nonlinear programming</a></li>

<li><a href="Entropic_value_at_risk" title="Entropic value at risk">Entropic value at risk</a></li></ul>

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

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<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFYudkowsky2008" class="citation web cs1">Yudkowsky, Eliezer (11 November 2008). <a rel="nofollow" class="external text" href="http://lesswrong.com/lw/vp/worse_than_random/">"Worse Than Random - LessWrong"</a>.</cite>

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<cite id="CITEREFGlover,_F.2007" class="citation journal cs1">Glover, F. (2007). "Tabu search—uncharted domains". Annals of Operations Research. 149: 89–98. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.417.8223">10.1.1.417.8223</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10479-006-0113-9">10.1007/s10479-006-0113-9</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:6854578">6854578</a>.</cite>

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<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>Michalewicz, Z. and Fogel, D. B. (2000), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=MpKqCAAAQBAJ&q=%22stochastic%22">How to Solve It: Modern Heuristics</a>, Springer-Verlag, New York.</li></ul>

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

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