Parabolic Relaxation for Quadratically-constrained Quadratic Programming -- Part I: Definitions & Basic Properties

TL;DR

This paper introduces a parabolic relaxation method for QCQP problems, which approximates the non-convex feasible set with convex quadratic constraints, enabling near-optimal solutions and potential acceleration of algorithms.

Contribution

It proposes a novel parabolic relaxation technique for QCQP, showing its properties, how to recover feasible points, and how to make it as strong as SDP relaxations with coordinate transformations.

Findings

Feasible set lies on the boundary of the relaxation.

Near-optimal feasible points can be recovered via penalization.

Parabolic relaxation can match SDP strength after coordinate change.

Abstract

For general quadratically-constrained quadratic programming (QCQP), we propose a parabolic relaxation described with convex quadratic constraints. An interesting property of the parabolic relaxation is that the original non-convex feasible set is contained on the boundary of the parabolic relaxation. Under certain assumptions, this property enables one to recover near-optimal feasible points via objective penalization. Moreover, through an appropriate change of coordinates that requires a one-time computation of an optimal basis, the easier-to-solve parabolic relaxation can be made as strong as a semidefinite programming (SDP) relaxation, which can be effective in accelerating algorithms that require solving a sequence of convex surrogates. The majority of theoretical and computational results are given in the next part of this work [57].