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<title>Pattern search (optimization)</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Pattern search (optimization)</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Pattern_search_(disambiguation)" class="mw-redirect mw-disambig" title="Pattern search (disambiguation)">Pattern search</a>.</div>

<p><b>Pattern search</b> (also known as direct search, derivative-free search, or black-box search) is a family of numerical <a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">optimization</a> methods that does not require a <a href="Gradient" title="Gradient">gradient</a>. As a result, it can be used on functions that are not <a href="Continuous_function" title="Continuous function">continuous</a> or <a href="Differentiable_function" title="Differentiable function">differentiable</a>. One such pattern search method is "convergence" (see below), which is based on the theory of positive bases. Optimization attempts to find the best match (the solution that has the lowest error value) in a <a href="Multidimensional_analysis" title="Multidimensional analysis">multidimensional analysis</a> space of possibilities.

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

<p>The name "pattern search" was coined by Hooke and Jeeves.<sup id="cite_ref-hooke61direct_1-0" class="reference"><a href="#cite_note-hooke61direct-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> An early and simple variant is attributed to <a href="Enrico_Fermi" title="Enrico Fermi">Fermi</a> and <a href="Nicholas_Metropolis" title="Nicholas Metropolis">Metropolis</a> when they worked at the <a href="Los_Alamos_National_Laboratory" title="Los Alamos National Laboratory">Los Alamos National Laboratory</a>. It is described by Davidon,<sup id="cite_ref-davidon91variable_2-0" class="reference"><a href="#cite_note-davidon91variable-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> as follows:

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</style><blockquote class="templatequote"><p>They varied one theoretical parameter at a time by steps of the same magnitude, and when no such increase or decrease in any one parameter further improved the fit to the experimental data, they halved the step size and repeated the process until the steps were deemed sufficiently small.</p></blockquote>

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

<p>Convergence is a pattern search method proposed by Yu, who proved that it converges using the theory of positive bases.<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> Later, <a href="Virginia_Torczon" title="Virginia Torczon">Torczon</a>, Lagarias and co-authors<sup id="cite_ref-torczon97convergence_4-0" class="reference"><a href="#cite_note-torczon97convergence-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-dolan03convergence_5-0" class="reference"><a href="#cite_note-dolan03convergence-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> used positive-basis techniques to prove the convergence of another pattern-search method on specific classes of functions. Outside of such classes, pattern search is a <a href="Heuristic_(computer_science)" title="Heuristic (computer science)">heuristic</a> that can provide useful approximate solutions for some issues, but can fail on others. Outside of such classes, pattern search is not an <a href="Iterative_method" title="Iterative method">iterative method</a> that converges to a solution; indeed, pattern-search methods can converge to non-stationary points on some relatively tame problems.<sup id="cite_ref-PM_6-0" class="reference"><a href="#cite_note-PM-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-McKnnon99_7-0" class="reference"><a href="#cite_note-McKnnon99-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

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<ul><li><a href="Golden-section_search" title="Golden-section search">Golden-section search</a> conceptually resembles PS in its narrowing of the search range, only for single-dimensional search spaces.</li>

<li><a href="Nelder%E2%80%93Mead_method" title="Nelder–Mead method">Nelder–Mead method</a> aka. the simplex method conceptually resembles PS in its narrowing of the search range for multi-dimensional search spaces but does so by maintaining <i>n</i> + 1 points for <i>n</i>-dimensional search spaces, whereas PS methods computes 2<i>n</i> + 1 points (the central point and 2 points in each dimension).</li>

<li><a href="Luus%E2%80%93Jaakola" title="Luus–Jaakola">Luus–Jaakola</a> samples from a <a href="Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a> surrounding the current position and uses a simple formula for exponentially decreasing the sampling range.</li>

<li><a href="Random_search" title="Random search">Random search</a> is a related family of optimization methods that sample from a <a href="Hypersphere" class="mw-redirect" title="Hypersphere">hypersphere</a> surrounding the current position.</li>

<li><a href="Random_optimization" title="Random optimization">Random optimization</a> is a related family of optimization methods that sample from a <a href="Normal_distribution" title="Normal distribution">normal distribution</a> surrounding the current position.</li></ul>

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

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</style><cite id="CITEREFHookeJeeves1961" class="citation journal cs1">Hooke, R.; Jeeves, T.A. (1961). <a rel="nofollow" class="external text" href="https://doi.org/10.1145%2F321062.321069">""Direct search" solution of numerical and statistical problems"</a>. <i>Journal of the ACM</i>. <b>8</b> (2): <span class="nowrap">212–</span>229. <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/10.1145%2F321062.321069">10.1145/321062.321069</a></span>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10905054">10905054</a>.</cite></span>

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<cite id="CITEREFDavidon1991" class="citation journal cs1">Davidon, W.C. (1991). "Variable metric method for minimization". <i>SIAM Journal on Optimization</i>. <b>1</b> (1): <span class="nowrap">1–</span>17. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.693.272">10.1.1.693.272</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F0801001">10.1137/0801001</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:1819475">1819475</a>.</cite></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">*Yu, Wen Ci. 1979. “<a rel="nofollow" class="external text" href="http://engine.scichina.com/doi/10.1360/za1979-9-S1-53">Positive basis and a class of direct search techniques</a>”. <i>Scientia Sinica</i> [<i>Zhongguo Kexue</i>]: 53—68.

<ul><li>Yu, Wen Ci. 1979. “<a rel="nofollow" class="external text" href="http://engine.scichina.com/doi/10.1360/za1979-9-S1-69?scroll=true">The convergent property of the simplex evolutionary technique</a>”. <i>Scientia Sinica</i> [<i>Zhongguo Kexue</i>]: 69–77.</li></ul>

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<cite id="CITEREFTorczon1997" class="citation journal cs1"><a href="Virginia_Torczon" title="Virginia Torczon">Torczon, V.J.</a> (1997). <a rel="nofollow" class="external text" href="http://www.cs.wm.edu/~va/research/unc.pdf">"On the convergence of pattern search algorithms"</a> <span class="cs1-format">(PDF)</span>. <i>SIAM Journal on Optimization</i>. <b>7</b> (1): <span class="nowrap">1–</span>25. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.3173">10.1.1.50.3173</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS1052623493250780">10.1137/S1052623493250780</a>.</cite></span>

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<cite id="CITEREFDolanLewisTorczon2003" class="citation journal cs1">Dolan, E.D.; Lewis, R.M.; Torczon, V.J. (2003). <a rel="nofollow" class="external text" href="http://www.cs.wm.edu/~va/research/local.pdf">"On the local convergence of pattern search"</a> <span class="cs1-format">(PDF)</span>. <i>SIAM Journal on Optimization</i>. <b>14</b> (2): <span class="nowrap">567–</span>583. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.78.2407">10.1.1.78.2407</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS1052623400374495">10.1137/S1052623400374495</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/2060%2F20000109966">2060/20000109966</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:4226940">4226940</a>.</cite></span>

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<li id="cite_note-PM-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-PM_6-0">^</a></b></span> <span class="reference-text">* <a href="Michael_J._D._Powell" title="Michael J. D. Powell">Powell, Michael J. D.</a> 1973. ”<a rel="nofollow" class="external text" href="https://link.springer.com/article/10.1007/BF01584660">On Search Directions for Minimization Algorithms</a>.” <i>Mathematical Programming</i> 4: 193—201.</span>

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<li id="cite_note-McKnnon99-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-McKnnon99_7-0">^</a></b></span> <span class="reference-text">* <cite id="CITEREFMcKinnon1999" class="citation journal cs1">McKinnon, K. I. M. (1999). "Convergence of the Nelder–Mead simplex method to a non-stationary point". <i>SIAM J. Optim</i>. <b>9</b>: <span class="nowrap">148–</span>158. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.3900">10.1.1.52.3900</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS1052623496303482">10.1137/S1052623496303482</a>.</cite> (algorithm summary online).</span>

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</style><div id="Major_subfields_of_optimization78" style="font-size:114%;margin:0 4em"><a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">Major subfields of optimization</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Convex_programming" class="mw-redirect" title="Convex programming">Convex programming</a></li>

<li><a href="Fractional_programming" title="Fractional programming">Fractional programming</a></li>

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

<li><a href="Quadratic_programming" title="Quadratic programming">Quadratic programming</a></li>

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

<li><a href="Stochastic_programming" title="Stochastic programming">Stochastic programming</a></li>

<li><a href="Robust_optimization" title="Robust optimization">Robust optimization</a></li>

<li><a href="Combinatorial_optimization" title="Combinatorial optimization">Combinatorial optimization</a></li>

<li><a href="Infinite-dimensional_optimization" title="Infinite-dimensional optimization">Infinite-dimensional optimization</a></li>

<li><a href="Metaheuristic" title="Metaheuristic">Metaheuristics</a></li>

<li><a href="Constraint_satisfaction" title="Constraint satisfaction">Constraint satisfaction</a></li>

<li><a href="Multiobjective_optimization" class="mw-redirect" title="Multiobjective optimization">Multiobjective optimization</a></li>

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

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