Stability analysis of stochastic integer optimization problems

This paper presents the stability analysis of integer linear programs with respect to perturbations in stochastic data, namely Markov chains. These perturbations affect the initial distribution, the transition matrix, or the stationary distribution of Markov chains. Stability analysis is concerned with obtaining the set of all perturbations for which a solution to the integer optimization problem remains optimal. In particular, we derive expressions for stability regions for perturbations in the initial distribution, the transition matrix and the stationary distribution. The constraints that preserve the stochasticity of the problem data are affine. The intersection of the stability region for arbitrary perturbations with these affine constraints yields the desired stability regions. Finally, stability regions for perturbations of elements of the transition matrix, given that the problem is linear in the stationary distribution of that transition matrix, are obtained using a small perturbation analysis. The results are applied to sensor placement problems, where a phenomenon modeled as a Markov chain needs to be detected, and numerical examples are given.