Intersection disjunctions for reverse convex sets

We present a framework to obtain valid inequalities for optimization problems constrained by a reverse convex set, which is defined as the set of points in a polyhedron that lie outside a given open convex set. We are particularly interested in cases where the closure of the convex set is either non-polyhedral, or is defined by too many inequalities to directly apply disjunctive programming. Reverse convex sets arise in many models, including bilevel optimization and polynomial optimization. Intersection cuts are a well-known method for generating valid inequalities for a reverse convex set. Intersection cuts are generated from a basic solution that lies within the convex set. Our contribution is a framework for deriving valid inequalities for the reverse convex set from basic solutions that lie outside the convex set. We begin by proposing an extension to intersection cuts that defines a two-term disjunction for a reverse convex set. Next, we generalize this analysis to a multi-term disjunction by considering the convex set's recession directions. These disjunctions can be used in a cut-generating linear program to obtain disjunctive cuts for the reverse convex set.



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