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

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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 \(j \ne k\), 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-time-checkable 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.

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Notes

  1. In fact, if the number of inactive linear inequalities at \(Y^*\) is known ahead of time, then this bound can be improved. For example, suppose (2) contains the two inequalities \(0 \le X_{12} \le 1\). Then the rank bound can be improved to \(\lceil \sqrt{2m} \rceil \) since both inequalities cannot be active at the same time.

  2. Interestingly, compared to [19], this provides a simple proof that every polynomial optimization problem has a corresponding SDP relaxation in which \(r^* = 2\); see Sect. 3.

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Acknowledgements

We are in debt to the anonymous associate editor and two referees, who suggested many positive improvements to the paper. We would also like to thank Gang Luo, who pointed out an error in the knapsack example.

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Authors and Affiliations

  1. Department of Management Sciences, University of Iowa, Iowa City, IA, 52242-1994, USA

    Samuel Burer

  2. Department of Management Science and Engineering, Institute of Computational and Mathematical Engineering, Stanford University, Stanford, CA, 94305-4121, USA

    Yinyu Ye

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Correspondence to Samuel Burer.

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Burer, S., Ye, Y. Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs. Math. Program. 181, 1–17 (2020). https://doi.org/10.1007/s10107-019-01367-2

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