Quadratic programs with hollows

Let $$\mathcal {F}$$F be a quadratically constrained, possibly nonconvex, bounded set, and let $$\mathcal {E}_1, \ldots, \mathcal {E}_l$$E1,ý,El denote ellipsoids contained in $$\mathcal {F}$$F with non-intersecting interiors. We prove that minimizing an arbitrary quadratic $$q(\cdot )$$q(·) over $$\mathcal {G}:= \mathcal {F}{\setminus } \cup _{k=1}^\ell {{\mathrm{int}}}(\mathcal {E}_k)$$G:=F\ýk=1ℓint(Ek) is no more difficult than minimizing $$q(\cdot )$$q(·) over $$\mathcal {F}$$F in the following sense: if a given semidefinite-programming (SDP) relaxation for $$\min \{ q(x) : x \in \mathcal {F}\}$$min{q(x):xýF} is tight, then the addition of l linear constraints derived from $$\mathcal {E}_1, \ldots, \mathcal {E}_l$$E1,ý,El yields a tight SDP relaxation for $$\min \{ q(x) : x \in \mathcal {G}\}$$min{q(x):xýG}. We also prove that the convex hull of $$\{ (x,xx^T) : x \in \mathcal {G}\}$${(x,xxT):xýG} equals the intersection of the convex hull of $$\{ (x,xx^T) : x \in \mathcal {F}\}$${(x,xxT):xýF} 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.