 
Summary: A Sample Approximation Approach for
Optimization with Probabilistic Constraints
James Luedtke and Shabbir Ahmed
H. Milton Stewart School of Industrial & Systems Engineering
Georgia Institute of Technology, Atlanta, GA 30332
jluedtk@us.ibm.com, sahmed@isye.gatech.edu
September 17, 2007
Abstract
We study approximations of optimization problems with probabilistic constraints in which the original
distribution of the underlying random vector is replaced with an empirical distribution obtained from
a random sample. We show that such a sample approximation problem with risk level larger than the
required risk level will yield a lower bound to the true optimal value with probability approaching one
exponentially fast. This leads to an a priori estimate of the sample size required to have high confidence
that the sample approximation will yield a lower bound. We then provide conditions under which solving
a sample approximation problem with a risk level smaller than the required risk level will yield feasible
solutions to the original problem with high probability. Once again, we obtain a priori estimates on the
sample size required to obtain high confidence that the sample approximation problem will yield a feasible
solution to the original problem. Finally, we present numerical illustrations of how these results can be
used to obtain feasible solutions and optimality bounds for optimization problems with probabilistic
constraints.
