An Alternative Perspective on Copositive and Convex Relaxations of Nonconvex Quadratic Programs
Abstract
We study convex relaxations of nonconvex quadratic programs. We identify a family of socalled feasibility preserving convex relaxations, which includes the wellknown copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful properties of the corresponding convex underestimators. In particular, if the recession cone of the feasible region of the quadratic program does not contain any directions of negative curvature, we show that the convex underestimator arising from the copositive relaxation is precisely the convex envelope of the objective function of the quadratic program, providing another proof of Burer's wellknown result on the exactness of the copositive relaxation. We also present an algorithmic recipe for constructing instances of quadratic programs with a finite optimal value but an unbounded doubly nonnegative relaxation.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 DOI:
 10.48550/arXiv.2006.00301
 arXiv:
 arXiv:2006.00301
 Bibcode:
 2020arXiv200600301Y
 Keywords:

 Mathematics  Optimization and Control;
 90C20;
 90C25;
 90C26
 EPrint:
 26 pages