Stochastic Programming with Probability Constraints

Laetitia Andrieu (1) , Guy Cohen (2) , Felisa Vázquez-Abad (3)

1 EDF R&D OSIRIS - Optimisation, Simulation, Risque et Statistiques pour les Marchés de l’Energie
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
3 Dep. Math. Stat. U. Melbourne

In this work we study optimization problems subject to a failure constraint. This constraint is expressed in terms of a condition that causes failure, representing a physical or technical breakdown. We formulate the problem in terms of a probability constraint, where the level of "confidence" is a modelling parameter and has the interpretation that the probability of failure should not exceed that level. Application of the stochastic Arrow-Hurwicz algorithm poses two difficulties: one is structural and arises from the lack of convexity of the probability constraint, and the other is the estimation of the gradient of the probability constraint. We develop two gradient estimators with decreasing bias via a convolution method and a finite difference technique, respectively, and we provide a full analysis of convergence of the algorithms. Convergence results are used to tune the parameters of the numerical algorithms in order to achieve best convergence rates, and numerical results are included via an example of application in finance.

Mots clés: Probability constraints; stochastic programming; stochastic gradient algorithm; stochastic approximation

Domaines: Optimisation et contrôle [math.OC]

Laetitia Andrieu, Guy Cohen, Felisa Vázquez-Abad. Stochastic Programming with Probability Constraints. 2007. ⟨hal-00166149⟩