On the tightness of SDP relaxations of QCQPs

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.

Automated summary

This paper studies the conditions of SDP relaxation for QCQP problems, proving the tightness of SDP relaxation when the quadratic eigenvalue multiplicity is large enough, presenting similar sufficient conditions that the projected epigraph of SDP gives the convex hull of the original QCQP's epigraph. Our results also imply new sufficient conditions for the tightness and convex hull exactness of second order cone program relaxation of simultaneously diagonalizable QCQPs.