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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Random optimization</span></span>
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<p><b>Random optimization (RO)</b> is a family of numerical <a href="Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">optimization</a> methods <a href="Derivative-free_optimization" title="Derivative-free optimization">that do not require the gradient</a> of the optimization problem and RO can hence 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>. Such optimization methods are also known as direct-search, derivative-free, or black-box methods.
</p><p>The name random optimization is attributed to Matyas <sup id="cite_ref-matyas65random_1-0" class="reference"><a href="#cite_note-matyas65random-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> who made an early presentation of RO along with basic mathematical analysis. RO works by iteratively moving to better positions in the search-space which are sampled using e.g. a <a href="Normal_distribution" title="Normal distribution">normal distribution</a> surrounding the current position.
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<div class="mw-heading mw-heading2"><h2 id="Algorithm">Algorithm</h2></div>
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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Simulated_annealing" title="Simulated annealing">Simulated annealing</a></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }">
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<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }</annotation>
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</math></span><img src="./ab54a3a84df609b7ac88c7f37cc10cbd5e5761fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.404ex; height:2.676ex;" alt="{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }" loading="lazy"></span> be the fitness or cost function which must be minimized. Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in \mathbb {R} ^{n}}">
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<ul><li>Initialize <b>x</b> with a random position in the search-space.</li>
<li>Until a termination criterion is met (e.g. number of iterations performed, or adequate fitness reached), repeat the following:
<ul><li>Sample a new position <b>y</b> by adding a <a href="Normal_distribution" title="Normal distribution">normally distributed</a> random vector to the current position <b>x</b></li>
<li>If (<i>f</i>(<b>y</b>) < <i>f</i>(<b>x</b>)) then move to the new position by setting <b>x</b> = <b>y</b></li></ul></li>
<li>Now <b>x</b> holds the best-found position.</li></ul>
<p>This algorithm corresponds to a (1+1) <a href="Evolution_strategy" title="Evolution strategy">evolution strategy</a> with constant step-size.
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<div class="mw-heading mw-heading2"><h2 id="Convergence_and_variants">Convergence and variants</h2></div>
<p>Matyas showed the basic form of RO converges to the optimum of a simple <a href="Unimodal_function" class="mw-redirect" title="Unimodal function">unimodal function</a> by using a <a href="Limit_(mathematics)" title="Limit (mathematics)">limit-proof</a> which shows convergence to the optimum is certain to occur if a potentially infinite number of iterations are performed. However, this proof is not useful in practice because a finite number of iterations can only be executed. In fact, such a theoretical limit-proof will also show that purely random sampling of the search-space will inevitably yield a sample arbitrarily close to the optimum.
</p><p>Mathematical analyses are also conducted by Baba <sup id="cite_ref-baba81convergence_2-0" class="reference"><a href="#cite_note-baba81convergence-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and Solis and Wets <sup id="cite_ref-solis81random_3-0" class="reference"><a href="#cite_note-solis81random-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> to establish that convergence to a region surrounding the optimum is inevitable under some mild conditions for RO variants using other <a href="Probability_distribution" title="Probability distribution">probability distributions</a> for the sampling. An estimate on the number of iterations required to approach the optimum is derived by Dorea.<sup id="cite_ref-dorea83expected_4-0" class="reference"><a href="#cite_note-dorea83expected-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> These analyses are criticized through empirical experiments by Sarma <sup id="cite_ref-sarma90convergence_5-0" class="reference"><a href="#cite_note-sarma90convergence-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> who used the optimizer variants of Baba and Dorea on two real-world problems, showing the optimum to be approached very slowly and moreover that the methods were actually unable to locate a solution of adequate fitness, unless the process was started sufficiently close to the optimum to begin with.
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Random_search" title="Random search">Random search</a> is a closely related family of optimization methods which sample from a <a href="Hypersphere" class="mw-redirect" title="Hypersphere">hypersphere</a> instead of a normal distribution.</li>
<li><a href="Luus%E2%80%93Jaakola" title="Luus–Jaakola">Luus–Jaakola</a> is a closely related optimization method using a <a href="Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a> in its sampling and a simple formula for exponentially decreasing the sampling range.</li>
<li><a href="Pattern_search_(optimization)" title="Pattern search (optimization)">Pattern search</a> takes steps along the axes of the search-space using exponentially decreasing step sizes.</li>
<li><a href="Stochastic_optimization" title="Stochastic optimization">Stochastic optimization</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-matyas65random-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-matyas65random_1-0">^</a></b></span> <span class="reference-text">
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</style><cite id="CITEREFMatyas1965" class="citation journal cs1">Matyas, J. (1965). <a rel="nofollow" class="external text" href="https://www.mathnet.ru/eng/at11288">"Random optimization"</a>. <i>Automation and Remote Control</i>. <b>26</b> (2): <span class="nowrap">246–</span>253.</cite></span>
</li>
<li id="cite_note-baba81convergence-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-baba81convergence_2-0">^</a></b></span> <span class="reference-text">
<cite id="CITEREFBaba1981" class="citation journal cs1">Baba, N. (1981). "Convergence of a random optimization method for constrained optimization problems". <i>Journal of Optimization Theory and Applications</i>. <b>33</b> (4): <span class="nowrap">451–</span>461. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf00935752">10.1007/bf00935752</a>.</cite></span>
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<li id="cite_note-solis81random-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-solis81random_3-0">^</a></b></span> <span class="reference-text">
<cite id="CITEREFSolisWets1981" class="citation journal cs1">Solis, Francisco J.; <a href="Roger_J-B_Wets" title="Roger J-B Wets">Wets, Roger J.-B.</a> (1981). "Minimization by random search techniques". <i><a href="Mathematics_of_Operations_Research" title="Mathematics of Operations Research">Mathematics of Operations Research</a></i>. <b>6</b> (1): <span class="nowrap">19–</span>30. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1287%2Fmoor.6.1.19">10.1287/moor.6.1.19</a>.</cite></span>
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<li id="cite_note-dorea83expected-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-dorea83expected_4-0">^</a></b></span> <span class="reference-text">
<cite id="CITEREFDorea1983" class="citation journal cs1">Dorea, C.C.Y. (1983). "Expected number of steps of a random optimization method". <i>Journal of Optimization Theory and Applications</i>. <b>39</b> (3): <span class="nowrap">165–</span>171. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf00934526">10.1007/bf00934526</a>.</cite></span>
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<li id="cite_note-sarma90convergence-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-sarma90convergence_5-0">^</a></b></span> <span class="reference-text">
<cite id="CITEREFSarma1990" class="citation journal cs1">Sarma, M.S. (1990). "On the convergence of the Baba and Dorea random optimization methods". <i>Journal of Optimization Theory and Applications</i>. <b>66</b> (2): <span class="nowrap">337–</span>343. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf00939542">10.1007/bf00939542</a>.</cite></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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