Scalable Heuristics for a Class of Chance-Constrained Stochastic Programs
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ACM Transactions on Modeling and Computer Simulation (TOMACS)
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Easy distributions for combinatorial optimization problems with probabilistic constraints
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Discrete Optimization
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INFORMS Journal on Computing
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INFORMS Journal on Computing
Improved integer programming approaches for chance-constrained stochastic programming
IJCAI'13 Proceedings of the Twenty-Third international joint conference on Artificial Intelligence
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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 a 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.