On intersection of two mixing sets with applications to joint chance-constrained programs

Abstract

We study the polyhedral structure of a generalization of a mixing set described by the intersection of two mixing sets with two shared continuous variables, where one continuous variable has a positive coefficient in one mixing set, and a negative coefficient in the other. Our developments are motivated from a key substructure of linear joint chance-constrained programs (CCPs) with random right hand sides from a finite probability space. The CCPs of interest immediately admit a mixed-integer programming reformulation. Nevertheless, such standard reformulations are difficult to solve at large-scale due to the weakness of their linear programming relaxations. In this paper, we initiate a systemic polyhedral study of such joint CCPs by explicitly analyzing the system obtained from simultaneously considering two linear constraints inside the chance constraint. We carry out our study on this particular intersection of two mixing sets under a nonnegativity assumption on data. Mixing inequalities are immediately applicable to our set, yet they are not sufficient. Therefore, we propose a new class of valid inequalities in addition to the mixing inequalities, and establish conditions under which these inequalities are facet defining. Moreover, under certain additional assumptions, we prove that these new valid inequalities along with the classical mixing inequalities are sufficient in terms of providing the closed convex hull description of our set. We also show that linear optimization over our set is polynomial-time, and we independently give a (high-order) polynomial-time separation algorithm for the new inequalities. We complement our theoretical results with a computational study on the strength of the proposed inequalities. Our preliminary computational experiments with a fast heuristic separation approach demonstrate that our proposed inequalities are practically effective as well.