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# Probability Distribution Problems Concerning Stochastic Programming Problems

• András Prékopa
Part of the C.I.M.E. Summer Schools book series (CIME, volume 38)

## Abstract

Different kinds of stochastic programming models are formulated in the present mathematical programming literature. Their solutions lead to linear or non-linear deterministic programming problems. There are, however, a number of problems, mainly probability distribution problems, which remained unsolved which are nevertheless important and necessary to solve in order to be able to handle effectively these stochastic optimization problems. The main types of stochastic programming problems are the following.

a) Chance constrained programming. The problem is to minimize the expectation of a functional z(c, x) under the condition that
$${\text{P(g(A,}}\,{\text{x)}} \geqslant {\text{b)}} \geqslant \alpha$$
(1.1)
where g is a certain function of the elements of the matrix A and of the elements of the unknown vector x. ∝ is a prescribed probability usually near 1. In many practical cases the above problem reduces to the simpler form:
$$\min {\text{imize}}\,{\text{E}}\left( {{\text{c'}}\,{\text{x}}} \right)$$
subject to the condition
$${\text{P}}\left( {{\text{Ax}} \geqslant {\text{b}}} \right) \geqslant \alpha$$
(1.2)
The matrix A and the vector b are partly or entirely random. Among several papers where this problem is investigated we mention  and [2l].

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2010

## Authors and Affiliations

• András Prékopa
• 1
1. 1.Tulane-UniversityNew OrleansUSA