Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

Abstract

We study a class of quadratically constrained quadratic programs (QCQPs), called diagonal QCQPs, which contain no off-diagonal terms x j x k for j = k, and we provide a sufficient condition on the problem data guaranteeing that the basic Shor semidefinite relaxation is exact. Our condition complements and refines those already present in the literature and can be checked in polynomial time. We then extend our analysis from diagonal QCQPs to general QCQPs, i.e., ones with no particular structure. By reformulating a general QCQP into diagonal form, we establish new, polynomial-timecheckable sufficient conditions for the semidefinite relaxations of general QCQPs to be exact. Finally, these ideas are extended to show that a class of random general QCQPs has exact semidefinite relaxations with high probability as long as the number of constraints grows no faster than a fixed polynomial in the number of variables. To the best of our knowledge, this is the first result establishing the exactness of the semidefinite relaxation for random general QCQPs.

On Convex Hulls of Epigraphs of QCQPs

“…Recently, a number of exciting results in phase retrieval [17] and clustering [1,28,31] have shown that under various assumptions on the data, the QCQP formulation of the corresponding problem has a tight SDP relaxation. In contrast to these results, which address QCQPs arising from particular arXiv:2002.01566v1 [math.OC] 4 Feb 2020 problems, Burer and Ye [16] very recently gave appealing deterministic sufficient conditions under which the standard SDP relaxation of general QCQPs is tight. In our paper, we continue this vein of research for general QCQPs.…”

Correction to: Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs

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This note reports an error in Theorem 3 of [1], which was pointed about by Prof. Fatma Kılınç-Karzan and Mr. Alex Wang of Carnegie Mellon University, whom we sincerely thank for bringing this to our attention.The error can be seen in the proof of Theorem 3, where system ( 12) is analyzed for given diagonal matrices {C, B i , D i } and variable diagonal matrices {X , Y i }. In this setting, C, B i 0, while each D i is indefinite.

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“…This note reports an error in Theorem 3 of [1], which was pointed about by Prof. Fatma Kılınç-Karzan and Mr. Alex Wang of Carnegie Mellon University, whom we sincerely thank for bringing this to our attention.…”

“…where (t, b) ∈ R 1+m . This formulation ( ‡) emphasizes the separable nature of each quadratic function in (1). In addition to ( †), define K j := cone{ f k − f j : k = j} ⊆ R 1+m and L j := {λd j : λ ∈ R} ⊆ R 1+m for each j. K j is the cone of feasible directions at f j in the polytope Q := conv{ f 1 , .…”

A Lifting-Penalty Method for Quadratic Programming with a Quadratic Matrix Inequality Constraint

“…Rank-constrained optimization problems are considered in [28,29]. In [30], polynomial-time checkable sufficient conditions, which guarantees that the semidefinite relaxations of quadratically constrained quadratic programs are exact, are given.…”