Stochastic programming: Difference between revisions

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'''[[stochastics|Stochastic]] programming''' is a framework for modeling [[Optimization (mathematics)|optimization]] problems that involve [[uncertainty]]. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include some unknown parameters. When the parameters are known only within certain bounds, one approach to tackling such problems is called [[robust optimization]]. Here the goal is to find a solution which is feasible for all such data and [[Optimization (mathematics)|optimal]] in some sense. [[Stochastic]] programming [[mathematical model|models]] are similar in style but take advantage of the fact that [[probability distributions]] governing the data are known or can be estimated. The goal here is to find some policy that is feasible for all (or almost all) the possible data instances and maximizes the expectation of some function of the decisions and the random variables. More generally, such models are formulated, solved analytically or numerically, and analyzed in order to provide useful information to a decision-maker.<ref>{{cite book| last1=Shapiro|first1=Alexander|last2=Dentcheva|first2=Darinka|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 (SIAM)|___location=Philadelphia, PA| publisher2=Mathematical Programming Society (MPS)| year=2009|pages=xvi+436|isbn=978-0-898716-87-0| url=http://www2.isye.gatech.edu/people/faculty/Alex_Shapiro/SPbook.pdf | mr=2562798 }}</ref>

As an example, consider two-stage [[linear program]]s. Here the decision maker takes some action in the first stage, after which a random event occurs affecting the outcome of the first-stage decision. A recourse decision can then be made in the second stage that compensates for any bad effects that might have been experienced as a result of the first-stage decision. The optimal policy from such a model is a single first-stage policy and a collection of recourse decisions (a decision rule) defining which second-stage action should be taken in response to each random outcome.

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==Biological Applications==

Stochastic [[dynamic programming]] is frequently used to model [[animal behaviour]] in such fields as [[behavioural ecology]].<ref>Mangel, M. & Clark, C. W. 1988. ''Dynamic modeling in behavioral ecology.'' Princeton University Press ISBN 0-691-08506-4</ref><ref>Houston, A. I & McNamara, J. M. 1999. ''Models of adaptive behaviour: an approach based on state''. Cambridge University Press ISBN 0-521-65539-0</ref> Empirical tests of models of [[Optimal foraging theory|optimal foraging]], [[Biological life cycle|life-history]] transitions such as [[Fledge|fledging in birds]] and egg laying in [[parasitoid]] wasps have shown the value of this modelling technique in explaining the evolution of behavioural decision making. These models are typically many staged, rather than two-staged.

==Economic Applications==

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{{Reflist}}

==Further reading==

* John R. Birge and François V. Louveaux. ''Introduction to Stochastic Programming''. Springer Verlag, New York, 1997.

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==External links==

* [http://stoprog.org Stochastic Programming Community Home Page].

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