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{{For|the context of control theory|Stochastic control}}
In the field of [[mathematical optimization]], '''stochastic programming''' is a framework for [[Mathematical model|modeling]] [[Optimization (mathematics)|optimization]] problems that involve [[uncertainty]]. A '''stochastic program''' is an optimization problem in which some or all problem parameters are uncertain, but follow known [[probability distribution]]s.<ref>{{cite book|last1=Shapiro|first1=Alexander|author2-link=Darinka Dentcheva|last2=Dentcheva|first2=Darinka|author3-link=Andrzej Piotr Ruszczyński|last3=Ruszczyński|first3=Andrzej|title=Lectures on stochastic programming: Modeling and theory|series=MPS/SIAM Series on Optimization|volume=9|publisher=Society for Industrial and Applied Mathematics and the Mathematical Programming Society|___location=Philadelphia, PA|year=2009|pages=xvi+436|isbn=978-0-89871-687-0|url=http://www2.isye.gatech.edu/people/faculty/Alex_Shapiro/SPbook.pdf|mr=2562798|access-date=2010-09-22|archive-date=2020-03-24|archive-url=https://web.archive.org/web/20200324131907/https://www2.isye.gatech.edu/people/faculty/Alex_Shapiro/SPbook.pdf|url-status=dead}}</ref><ref>{{Cite book|last1=Birge|first1=John R.|last2=Louveaux|first2=François|date=2011|title=Introduction to Stochastic Programming|url=https://doi.org/10.1007/978-1-4614-0237-4|series=Springer Series in Operations Research and Financial Engineering|language=en-gb|doi=10.1007/978-1-4614-0237-4|isbn=978-1-4614-0236-7|issn=1431-8598}}</ref> This framework contrasts with deterministic optimization, in which all problem parameters are assumed to be known exactly. The goal of stochastic programming is to find a decision which both optimizes some criteria chosen by the decision maker, and appropriately accounts for the uncertainty of the problem parameters. Because many real-world decisions involve uncertainty, stochastic programming has found applications in a broad range of areas ranging from [[finance]] to [[transportation]] to energy optimization.<ref>
Stein W. Wallace and William T. Ziemba (eds.). ''[https://books.google.com/books?id=KAI0jsuyDPsC&q=%22Applications+of+Stochastic+Programming%22 Applications of Stochastic Programming]''. MPS-SIAM Book Series on Optimization 5, 2005.
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Several stochastic programming methods have been developed:
* Scenario-based methods including
* Stochastic integer programming for problems in which some variables must be integers
* [[Chance constrained programming]] for dealing with constraints that must be satisfied with a given probability
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Suppose <math>\xi</math> contains <math>d</math> independent random components, each of which has three possible realizations (for example, future realizations of each random parameters are classified as low, medium and high), then the total number of scenarios is <math>K=3^d</math>. Such ''exponential growth'' of the number of scenarios makes model development using expert opinion very difficult even for reasonable size <math>d</math>. The situation becomes even worse if some random components of <math>\xi</math> have continuous distributions.
====Monte Carlo sampling and
A common approach to reduce the scenario set to a manageable size is by using Monte Carlo simulation. Suppose the total number of scenarios is very large or even infinite. Suppose further that we can generate a sample <math>\xi^1,\xi^2,\dots,\xi^N</math> of <math>N</math> realizations of the random vector <math>\xi</math>. Usually the sample is assumed to be [[independent and identically distributed]] (i.i.d sample). Given a sample, the expectation function <math>q(x)=E[Q(x,\xi)]</math> is approximated by the sample average
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</math>
This formulation is known as the ''
=== Statistical inference ===
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