On the tightness of SDP relaxations of QCQPs

• 成果类型: Article
• 署名作者: Wang, Alex L.; Kilinc-Karzan, Fatma
• 署名单位: Carnegie Mellon University
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-020-01589-9
• 发表日期: 2022
• 页码: 33-73
• 关键词: semidefinite programming relaxation; quadratic optimization; algorithm; diagonalization; constraints; symmetry

摘要

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.