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On semidefinite descriptions for convex hulls of quadratic programs (2403.04752v2)
Published 7 Mar 2024 in math.OC
Abstract: 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 semidefinite program (SDP) relaxation. In this paper we study when the convex hull of the epigraph of a QCQP coincides with the projected epigraph of the SDP relaxation. We present a sufficient condition for convex hull exactness and show that this condition is further necessary under an additional geometric assumption. The sufficient condition is based on geometric properties of $\Gamma$, the cone of convex Lagrange multipliers, and its relatives $\Gamma_1$ and $\Gamma\circ$.