An Alternative Perspective on Copositive and Convex Relaxations of Nonconvex Quadratic Programs

TL;DR

This paper offers a new perspective on convex relaxations of nonconvex quadratic programs, highlighting their role as convex underestimators and establishing conditions for their exactness, with implications for optimization theory.

Contribution

It introduces a family of feasibility-preserving convex relaxations, including copositive and doubly nonnegative relaxations, and links them to convex underestimators of the objective function.

Findings

Copositive relaxation yields the convex envelope of the objective under certain conditions.

The family of relaxations is feasibility-preserving and includes well-known relaxations.

An algorithmic method is provided to construct instances with finite optimal value but unbounded relaxations.

Abstract

We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known 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 well-known 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.