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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|
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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A stochastic [[linear program]] is a specific instance of the classical two-stage stochastic program. A stochastic LP is built from a collection of multi-period linear programs (LPs), each having the same structure but somewhat different data. The <math>k^{th}</math> two-period LP, representing the <math>k^{th}</math> scenario, may be regarded as having the following form:
<math>
\begin{array}{lccccccc}
\text{Minimize} & f^T x & + & g^T y & + & h_k^Tz_k & & \\
\text{subject to} & Tx & + & Uy & & & = & r \\
& & & V_k y & + & W_kz_k & = & s_k \\
& x & , & y & , & z_k & \geq & 0
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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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* [[András Prékopa]]. Stochastic Programming. Kluwer Academic Publishers, Dordrecht, 1995.
* [[Andrzej Piotr Ruszczyński|Andrzej Ruszczynski]] and Alexander Shapiro (eds.) (2003) ''Stochastic Programming''. Handbooks in Operations Research and Management Science, Vol. 10, Elsevier.
* {{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
* Stein W. Wallace and William T. Ziemba (eds.) (2005) ''Applications of Stochastic Programming''. MPS-SIAM Book Series on Optimization 5
* {{cite book | last1=King|first1=Alan J.|last2=Wallace|first2=Stein W.| title=Modeling with Stochastic Programming | series=Springer Series in Operations Research and Financial Engineering| publisher=Springer| ___location=New York|year=2012|isbn=978-0-387-87816-4| url=https://www.springer.com/mathematics/probability/book/978-0-387-87816-4 }}
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