70 research outputs found
A Slightly Lifted Convex Relaxation for Nonconvex Quadratic Programming with Ball Constraints
Globally optimizing a nonconvex quadratic over the intersection of balls
in is known to be polynomial-time solvable for fixed .
Moreover, when , the standard semidefinite relaxation is exact. When
, it has been shown recently that an exact relaxation can be constructed
using a disjunctive semidefinite formulation based essentially on two copies of
the case. However, there is no known explicit, tractable, exact convex
representation for . In this paper, we construct a new, polynomially
sized semidefinite relaxation for all , which does not employ a disjunctive
approach. We show that our relaxation is exact for . Then, for ,
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 . Extending this construction: (i) we show that
nonconvex quadratic programming over 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
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