Quadratic Programs with Hollows

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=1 int(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, xx ) : x ∈ G} equals the intersection of the convex hull of {(x, xx ) : x ∈ F} with the same l linear constraints. Inspired by these results, we resolve a related question in a seemingly unrelated area, mixedinteger nonconvex quadratic programming.