Global Sensitivity Analysis via a Statistical Tolerance Approach

Sensitivity analysis and multiparametric programming in optimization modeling study variations of optimal value and solutions in the presence of uncertain input parameters. In this paper, we consider simultaneous variations in the inputs of the objective and constraint (jointly called the RIM parameters), where the uncertainty is represented as a multivariate probability distribution. We introduce a tolerance approach based on principal component analysis, which obtains a tolerance region that is suited to the given distribution and can be considered a confidence set for the random input parameters. Since a tolerance region may contain parameters with different optimal bases, we extend the tolerance approach to the case where multiple optimal bases cover the tolerance region, by studying theoretical properties of critical regions (defined as the set of input parameters having the same optimal basis). We also propose a computational algorithm to find critical regions covering a given tolerance region in the RIM parameter space. Our theoretical results on geometric properties of critical regions contribute to the existing theory of parametric programming with an emphasis on the case where RIM parameters vary jointly, and provide deeper geometric understanding of critical regions. We evaluate the proposed framework using a series of experiments for sensitivity analysis, for model predictive control of an inventory management problem, and for large optimization problem instances.