Quadratic programs with hollows

Abstract

Let F be a quadratically constrained, possibly nonconvex, bounded set, and let E1, … , El denote ellipsoids contained in F with non-intersecting interiors. We prove that minimizing an arbitrary quadratic q(·) over G:=F\∪k=1ℓint(Ek) is no more difficult than minimizing q(·) over F in the following sense: if a given semidefinite-programming (SDP) relaxation for min { q(x) : x∈ F} is tight, then the addition of l linear constraints derived from E1, … , El yields a tight SDP relaxation for min { q(x) : x∈ G}. We also prove that the convex hull of { (x, xxT) : x∈ G} equals the intersection of the convex hull of { (x, xxT) : x∈ F} with the same l linear constraints. Inspired by these results, we resolve a related question in a seemingly unrelated area, mixed-integer nonconvex quadratic programming.