On convex hulls of epigraphs 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 sufficient conditions for a convex hull result that immediately implies that 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 convex hull result holds whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. 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.Comment: To appear at IPCO 2020. This conference extended abstract covers material already online at arXiv:1911.0919

On obtaining the convex hull of quadratic inequalities via aggregations

A classical approach for obtaining valid inequalities for a set involves weighted aggregations of the inequalities that describe such set. When the set is described by linear inequalities, thanks to the Farkas lemma, we know that every valid inequality can be obtained using aggregations. When the inequalities describing the set are two quadratics, Yildiran showed that the convex hull of the set is given by at most two aggregated inequalities. In this work, we study the case of a set described by three or more quadratic inequalities. We show that, under technical assumptions, the convex hull of a set described by three quadratic inequalities can be obtained via (potentially infinitely many) aggregated inequalities. We also show, through counterexamples, that it is unlikely to have a similar result if either the technical conditions are relaxed, or if we consider four or more inequalities

A Geometric View of SDP Exactness in QCQPs and its Applications

Quadratically constrained quadratic programs (QCQPs) are a highly expressive class of nonconvex optimization problems. While QCQPs are NP-hard in general, they admit a natural convex relaxation via the standard (Shor) semidefinite program (SDP) relaxation. Towards understanding when this relaxation is exact, we study general QCQPs and their (projected) SDP relaxations. We present sufficient (and in some cases, also necessary) conditions for objective value exactness (the condition that the objective values of the QCQP and its SDP relaxation coincide) and convex hull exactness (the condition that the convex hull of the QCQP epigraph coincides with the epigraph of its SDP relaxation). Our conditions for exactness are based on geometric properties of $\Gamma$, the cone of convex Lagrange multipliers, and its relatives $\Gamma_P$ and $\Gamma^\circ$. These tools form the basis of our main message: questions of exactness can be treated systematically whenever $\Gamma$, $\Gamma_P$, or $\Gamma^\circ$ is well-understood. As further evidence of this message, we apply our tools to address questions of exactness for a prototypical QCQP involving a binary on-off constraint, quadratic matrix programs, the QCQP formulation of the partition problem, and random and semi-random QCQPs