Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs
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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.
KeywordsQuadratically constrained quadratic programming Semidefinite relaxation Low-rank solutions
Mathematics Subject Classification90C20 90C22 90C26
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.
- 7.Bhojanapalli, S., Boumal, N., Jain, P., Netrapalli, P.: Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form. In: Proceedings of Machine Learning Research. Presented at the 31st Conference on Learning Theory, vol 75, pp. 1–28Google Scholar
- 8.Bienstock, D., Michalka, A.: Polynomial solvability of variants of the trust-region subproblem. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 380–390Google Scholar
- 15.Diestel, R.: Graph Theory, Volume 173 of Graduate Texts in Mathematics, 5th edn. Springer, Berlin (2018)Google Scholar
- 18.Luo, Z.Q., Ma, W.K., So, A.M.C., Ye, Y., Zhang, S.: Semidefinite relaxation of quadratic optimization problems. IEEE Signal Process. Mag. 27(3), 20–34 (2010)Google Scholar
- 19.Madani, R., Fazelnia, G., Lavaei, J.: Rank-2 Matrix Solution for Semidefinite Relaxations of Arbitrary Polynomial Optimization Problems. Columbia University, New York (2014)Google Scholar
- 23.Shamsi, D., Taheri, N., Zhu, Z., Ye, Y.: Conditions for correct sensor network localization using SDP relaxation. In: Bezdek, K., Deza, A., Ye, Y. (eds.) Discrete Geometry and Optimization. Fields Institute Communications, vol. 69, pp. 279–301. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-319-00200-2_16
- 24.Shor, N.:. Quadratic optimization problems. Soviet J. Comput. Syst. Sci. 25, 1–11 (1987). Originally published in Tekhnicheskaya Kibernetika 1, 128–139 (1987)Google Scholar
- 26.So, A.M.C.: Probabilistic analysis of the semidefinite relaxation detector in digital communications. In: Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 698–711. SIAM, Philadelphia, PA (2010)Google Scholar