Quadratic programs with hollows

Abstract

Let $\mathcal {F}$ be a quadratically constrained, possibly nonconvex, bounded set, and let $\mathcal {E}_1, \ldots , \mathcal {E}_l$ denote ellipsoids contained in $\mathcal {F}$ with non-intersecting interiors. We prove that minimizing an arbitrary quadratic $q(\cdot )$ over $\mathcal {G}:= \mathcal {F}{\setminus } \cup _{k=1}^\ell {{\mathrm{int}}}(\mathcal {E}_k)$ is no more difficult than minimizing $q(\cdot )$ over $\mathcal {F}$ in the following sense: if a given semidefinite-programming (SDP) relaxation for $\min \{ q(x) : x \in \mathcal {F}\}$ is tight, then the addition of l linear constraints derived from $\mathcal {E}_1, \ldots , \mathcal {E}_l$ yields a tight SDP relaxation for $\min \{ q(x) : x \in \mathcal {G}\}$. We also prove that the convex hull of $\{ (x,xx^T) : x \in \mathcal {G}\}$ equals the intersection of the convex hull of $\{ (x,xx^T) : x \in \mathcal {F}\}$ 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.