Structural properties of linear probabilistic constraints

Abstract

The article provides a structural analysis of the feasible set defined by linear probabilistic constraints. Emphasis is laid on single (individual) probabilistic constraints. A classical convexity result by Van de Panne/Popp and Kataoka is extended to a broader class of distributions and to more general functions of the decision vector. The range of probability levels for which convexity can be expected is exactly identified. Apart from convexity, also nontriviality and compactness of the feasible set are precisely characterized at the same time. The relation between feasible sets with negative and with nonnegative right-hand side is revealed. Finally, an existence result is formulated for the more difficult case of joint probabilistic constraints.

This work was supported by the DFG Research Center MATHEON Mathematics for key technologies in Berlin.

Keywords:

• Probabilistic constraints
• Stochastic programming
• Chance constraints
• Stochastic optimization

Keywords:

• 90C15

Acknowledgement

The author wishes to thank Prof. A. Seeger (University of Avignon) for proposing, an improved version of Proposition 2.1.

Notes

This work was supported by the DFG Research Center MATHEON Mathematics for key technologies in Berlin.