On the tightness of SDP relaxations of QCQPs

TL;DR

This paper investigates conditions ensuring the tightness of SDP relaxations for QCQPs, revealing that high symmetry, measured by quadratic eigenvalue multiplicity, guarantees relaxation tightness and convex hull exactness.

Contribution

The paper introduces a general framework for establishing sufficient conditions for SDP relaxation tightness in QCQPs, emphasizing the role of problem symmetry.

Findings

SDP relaxation is tight when quadratic eigenvalue multiplicity is large.

Projected epigraph of SDP matches the convex hull of the original QCQP.

New conditions for tightness of second order cone relaxations in diagonalizable QCQPs.

Abstract

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the projected epigraph of the SDP gives the convex hull of the epigraph in the original QCQP. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.