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 $\|x\| \le \min \{ 1, g + h^T x \}$ 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.
A Slightly Lifted Convex Relaxation for Nonconvex Quadratic Programming with Ball Constraints
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