Stochastic Programming with Probability

TL;DR

This paper investigates optimization problems with failure probability constraints, developing gradient estimation methods and analyzing convergence, with applications demonstrated in finance.

Contribution

It introduces two new gradient estimators for probability constraints and provides convergence analysis for stochastic algorithms addressing non-convex failure constraints.

Findings

Developed convolution and finite difference gradient estimators.

Proved convergence of the proposed stochastic algorithms.

Demonstrated effectiveness with a financial application.

Abstract

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.