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{{Short description|Mathematical optimization approach}}
'''Chance Constrained Programming (CCP)''' is a [[mathematical optimization]] approach used to handle problems under uncertainty. It was first introduced by [[Abraham Charnes|Charnes]] and [[William W. Cooper|Cooper]] in 1959 and further developed by Miller and Wagner in 1965.<ref>{{cite journal |last1=Charnes |first1=Abraham |last2=Cooper |first2=William W. |title=Chance-Constrained Programming |journal=Management Science |date=1959 |volume=6 |issue=1 |pages=73–79 |doi=10.1287/mnsc.6.1.73}}</ref><ref>{{cite journal |last1=Miller |first1=L. R. |last2=Wagner |first2=H. M. |title=Chance-constrained programming with joint constraints |journal=Operations Research |date=1965 |volume=13 |issue=6 |pages=930–945 |doi=10.1287/opre.13.6.930}}</ref> CCP is widely used in various fields, including [[finance]], [[engineering]], and [[operations research]], to optimize decision-making processes where certain constraints need to be satisfied with a specified probability.
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== Solution Approaches ==
To solve CCP problems, the [[stochastic optimization]] problem is often relaxed into an equivalent deterministic problem. There are different approaches depending on the nature of the problem:
* '''Linear CCP''': For linear systems, the feasible region is typically convex, and the problem can be solved using [[linear programming]] techniques.
* '''Nonlinear CCP''': For nonlinear systems, the main challenge lies in computing the probabilities and their gradients. These problems often require [[nonlinear programming]] solvers.
* '''Dynamic Systems''': Dynamic systems involve time-dependent uncertainties, and the solution approach must account for the [[propagation of uncertainty]] over time.<ref name=pu/>
== Practical Applications ==
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