Stochastic programming: Difference between revisions

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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|url=http://www2.isye.gatech.edu/people/faculty/Alex_Shapiro/SPbook.pdf|title=Lectures on stochastic programming: Modeling and theory|last2=Dentcheva|first2=Darinka|last3=Ruszczyński|first3=Andrzej|publisher=Society for Industrial and Applied Mathematics (SIAM)|year=2009|isbn=978-0-89871-687-0|series=MPS/SIAM Series on Optimization|volume=9|___location=Philadelphia, PA|pages=xvi+436|mr=2562798|author2-link=Darinka Dentcheva|author3-link=Andrzej Piotr Ruszczyński|agency=Mathematical Programming Society (MPS)}}</ref><ref>{{Cite journalbook|lastlast1=Birge|firstfirst1=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&printsec=frontcover&dq=%22Applications+of+Stochastic+Programming%22&hl=en&sa=X&ved=0ahUKEwivt-nn2OfiAhURXa0KHYJMC9UQ6AEIKjAA#v=onepage&q=%22Applications%20of%20Stochastic%20Programming%22&f=false Applications of Stochastic Programming]''. MPS-SIAM Book Series on Optimization 5, 2005.

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# How to construct scenarios, see {{Section link||Scenario construction}};

# How to solve the deterministic equivalent. Optimizers such as [[CPLEX]], [[GNU Linear Programming Kit|GLPK]] and [[Gurobi]] can solve large linear/nonlinear problems. The NEOS Server,<ref name="neos">{{Cite web|url=http://www.neos-server.org/neos/|title = NEOS Server for Optimization}}</ref> hosted at the [[University of Wisconsin, Madison]], allows free access to many modern solvers. The structure of a deterministic equivalent is particularly amenable to apply decomposition methods,<ref>{{cite book|first2=Alexander|last2=Shapiro|last1=Ruszczyński|first1=Andrzej|title=Stochastic Programming|publisher=[[Elsevier]]|year=2003|isbn=978-0444508546|series=Handbooks in Operations Research and Management Science|volume=10|___location=Philadelphia|pages=700|author1-link=Andrzej Piotr Ruszczyński}}</ref> such as [[Benders' decomposition]] or scenario decomposition;

# How to measure quality of the obtained solution with respect to the "true" optimum.