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arXiv · 2403.04752

On semidefinite descriptions for convex hulls of quadratic programs

Alex L. Wang·Fatma Kilinc-Karzan
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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 $\Gamma^\circ$.

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Alex L. Wang, Fatma Kilinc-Karzan. 2024-03-07. On semidefinite descriptions for convex hulls of quadratic programs. https://arxiv.org/abs/2403.04752

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