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Solving Chance-Constrained Stochastic Programs via Sampling and Integer Programming

Summary: Solving Chance-Constrained Stochastic Programs via
Sampling and Integer Programming
Shabbir Ahmed and Alexander Shapiro
H. Milton Stewart School of Industrial & Systems Engineering
Georgia Institute of Technology, Atlanta, GA 30332
June 6, 2008
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 candi-
date 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 fi-
nite distribution of modest support, and is an NP-hard combinatorial optimization
problem. We adopt integer programming based methods for its solution. In partic-


Source: Ahmed, Shabbir - School of Industrial and Systems Engineering, Georgia Institute of Technology


Collections: Engineering