On the Quantile Cut Closure of Chance-Constrained Problems

A chance constrained problem involves a set of scenario constraints from which a small subset can be violated. Existing works typically consider a mixed integer programming (MIP) formulation of this problem by introducing binary variables to indicate which constraint systems are to be satisfied or violated. A variety of cutting plane approaches for this MIP formulation have been developed. In this paper we consider a family of cuts for chance constrained problems in the original space rather than those in the extended space of the MIP reformulation. These cuts, known as quantile cuts, can be viewed as a projection of the well known family of mixing inequalities for the MIP reformulation, onto the original problem space. We show the following results regarding quantile cuts: (i) the closure of all quantile cuts is a polyhedral set; (ii) separation of quantile cuts is in general NP-hard; (iii) successive application of quantile cut closures achieves the convex hull of the chance constrained problem in the limit; and (iv) in the pure integer setting this convergence is finite.