A slightly lifted convex relaxation for nonconvex quadratic programming with ball constraints

Abstract

Globally optimizing a nonconvex quadratic over the intersection of m balls in \(\mathbb {R}^n\) is known to be polynomial-time solvable for fixed m. Moreover, when \(m=1\), the standard semidefinite relaxation is exact. When \(m=2\), it has been shown recently that an exact relaxation can be constructed using a disjunctive semidefinite formulation based essentially on two copies of the \(m=1\) case. However, there is no known explicit, tractable, exact convex representation for \(m \ge 3\). In this paper, we construct a new, polynomially sized semidefinite relaxation for all m, which does not employ a disjunctive approach. We show that our relaxation is exact for \(m=2\). Then, for \(m \ge 3\), we demonstrate empirically that it is fast and strong compared to existing relaxations. The key idea of the relaxation is a simple lifting of the original problem into dimension \(n\, +\, 1\). Extending this construction: (i) we show that nonconvex quadratic programming over \(\Vert x\Vert \le \min \{ 1, g + h^T x \}\) has an exact semidefinite representation; and (ii) we construct a new relaxation for quadratic programming over the intersection of two ellipsoids, which globally solves all instances of a benchmark collection from the literature.