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{{Short description|Optimization technique in mathematics}}
'''Random optimization (RO)''' is a family of numerical [[Optimization (mathematics)|optimization]] methods [[Derivative-free optimization|that do not require the gradient]] of the optimization problem and RO can hence be used on functions that are not [[Continuous function|continuous]] or [[Differentiable function|differentiable]]. Such optimization methods are also known as direct-search, derivative-free, or black-box methods.

The name random optimization is attributed to Matyas <ref name=matyas65random/> 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 [[normal distribution]] surrounding the current position.

== Algorithm ==
{{See also|Simulated annealing}}
Let <math>f: \mathbb{R}^{n} \rarr \mathbb{R}</math> be the fitness or cost function which must be minimized. Let <math>x \isin \mathbb{R}^{n}</math> designate a position or candidate solution in the search-space. The basic RO algorithm can then be described as:

* Initialize '''x''' with a random position in the search-space.
* Until a termination criterion is met (e.g. number of iterations performed, or adequate fitness reached), repeat the following:
** Sample a new position '''y''' by adding a [[normal distribution|normally distributed]] random vector to the current position '''x'''
** If (''f''('''y''') < ''f''('''x''')) then move to the new position by setting '''x''' = '''y'''
* Now '''x''' holds the best-found position.

This algorithm corresponds to a (1+1) [[evolution strategy]] with constant step-size.

== Convergence and variants ==
Matyas showed the basic form of RO converges to the optimum of a simple [[unimodal function]] by using a [[Limit (mathematics)|limit-proof]] 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.

Mathematical analyses are also conducted by Baba <ref name=baba81convergence/> and Solis and Wets <ref name=solis81random/> to establish that convergence to a region surrounding the optimum is inevitable under some mild conditions for RO variants using other [[probability distribution]]s for the sampling. An estimate on the number of iterations required to approach the optimum is derived by Dorea.<ref name=dorea83expected/> These analyses are criticized through empirical experiments by Sarma <ref name=sarma90convergence/> 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.

== See also ==
* [[Random search]] is a closely related family of optimization methods which sample from a [[hypersphere]] instead of a normal distribution.
* [[Luus–Jaakola]] is a closely related optimization method using a [[Uniform distribution (continuous)|uniform distribution]] in its sampling and a simple formula for exponentially decreasing the sampling range.
* [[Pattern search (optimization)|Pattern search]] takes steps along the axes of the search-space using exponentially decreasing step sizes.
* [[Stochastic optimization]]

== References ==
{{reflist|refs=
<ref name=matyas65random>
{{cite journal
|last=Matyas
|first=J.
|title=Random optimization
|journal=Automation and Remote Control
|year=1965
|volume=26
|number=2
|pages=246–253
|url=http://www.mathnet.ru/eng/at11288
}}
</ref>

<ref name=baba81convergence>
{{cite journal
|last=Baba
|first=N.
|title=Convergence of a random optimization method for constrained optimization problems
|journal=Journal of Optimization Theory and Applications
|year=1981
|volume=33
|number=4
|pages=451–461
|doi=10.1007/bf00935752
}}
</ref>

<ref name=solis81random>
{{cite journal
| last1=Solis | first1=Francisco J.
| last2=Wets | first2=Roger J.-B. | authorlink2=Roger J-B Wets
| title=Minimization by random search techniques
| journal=[[Mathematics of Operations Research]]
| year=1981
| volume=6
| number=1
| pages=19–30
| doi=10.1287/moor.6.1.19}}
</ref>

<ref name=dorea83expected>
{{cite journal
|last1=Dorea
|first1=C.C.Y.
|title=Expected number of steps of a random optimization method
|journal=Journal of Optimization Theory and Applications
|year=1983
|volume=39
|number=3
|pages=165–171
|doi=10.1007/bf00934526
}}
</ref>

<ref name=sarma90convergence>
{{cite journal
|last1=Sarma
|first1=M.S.
|title=On the convergence of the Baba and Dorea random optimization methods
|journal=Journal of Optimization Theory and Applications
|year=1990
|volume=66
|number=2
|pages=337–343
|doi=10.1007/bf00939542
}}
</ref>
}}

{{Major subfields of optimization}}

{{DEFAULTSORT:Random Optimization}}
[[Category:Optimization algorithms and methods]]

Random optimization - Revision history

imported>Ranjan Nahata: /* top */Fixed grammatical error