Solving Chance-Constrained Stochastic Programs via Sampling and Integer Programming

Various applications in reliability and risk management give rise to optimization problems with constraints involving random parameters, which are required to be satisfied with a pre-specified probability threshold. There are two main difficulties with such chance-constrained problems. First, checking feasibility of a given candidate solution exactly is, in general, impossible since this requires evaluating quantiles of random functions. Second, the feasible region induced by chance constraints is, in general, non-convex leading to severe optimization challenges. In this tutorial we discuss an approach based on solving approximating problems using Monte Carlo samples of the random data. This scheme can be used to yield both feasible solutions and statistical optimality bounds with high confidence using modest sample sizes. The approximating problem is itself a chance-constrained problem, albeit with a finite distribution of modest support, and is an NP-hard combinatorial optimization problem. We adopt integer programming based methods for its solution. In particular, we discuss a family valid inequalities for a integer programming formulations for a special but large class of chance-constraint problems that have demonstrated significant computational advantages.