Postoptimality for mean‐risk stochastic mixed‐integer programs and its application

The mean‐risk stochastic mixed‐integer programs can better model complex decision problems under uncertainty than usual stochastic (integer) programming models. In order to derive theoretical results in a numerically tractable way, the contamination technique is adopted in this paper for the postoptimality analysis of the mean‐risk models with respect to changes in the scenario set, here the risk is measured by the lower partial moment. We first study the continuity of the objective function and the differentiability, with respect to the parameter contained in the contaminated distribution, of the optimal value function of the mean‐risk model when the recourse cost vector, the technology matrix and the right‐hand side vector in the second stage problem are all random. The postoptimality conclusions of the model are then established. The obtained results are applied to two‐stage stochastic mixed‐integer programs with risk objectives where the objective function is nonlinear with respect to the probability distribution. The current postoptimality results for stochastic programs are improved.