Advertisement

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

  • Samuel BurerEmail author
  • Yinyu Ye
Full Length Paper Series A

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.

Keywords

Quadratically constrained quadratic programming Semidefinite relaxation Low-rank solutions 

Mathematics Subject Classification

90C20 90C22 90C26 

Notes

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.

References

  1. 1.
    Adler, I., Megiddo, N.: A simplex algorithm whose average number of steps is bounded between two quadratic functions of the smaller dimension. J. Assoc. Comput. Mach. 32(4), 871–895 (1985)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anjos, M.F., Lasserre, J.B. (eds.): Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research and Management Science, vol. 166. Springer, New York (2012)zbMATHGoogle Scholar
  3. 3.
    Anstreicher, K.M., Ji, J., Potra, F.A., Ye, Y.: Probabilistic analysis of an infeasible-interior-point algorithm for linear programming. Math. Oper. Res. 24(1), 176–192 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bandeira, A.S., Boumal, N., Singer, A.: Tightness of the maximum likelihood semidefinite relaxation for angular synchronization. Math. Program. 163(1–2, Ser. A), 145–167 (2017)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Barvinok, A.: Problems of distance geometry and convex properties of quadratic maps. Discrete Comput. Geom. 13, 189–202 (1995)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Beck, A., Pan, D.: A branch and bound algorithm for nonconvex quadratic optimization with ball and linear constraints. J. Global Optim. 69(2), 309–342 (2017)MathSciNetzbMATHGoogle Scholar
  7. 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. 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
  9. 9.
    Borgwardt, K.H.: The Simplex Method—A Probabilistic Approach. Springer, New York (1987)zbMATHGoogle Scholar
  10. 10.
    Burer, S.: A gentle, geometric introduction to copositive optimization. Math. Program. 151(1, Ser. B), 89–116 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Burer, S., Anstreicher, K.M.: Second-order-cone constraints for extended trust-region subproblems. SIAM J. Optim. 23(1), 432–451 (2013)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Burer, S., Monteiro, R.D.C.: A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. 95(2, Ser. B), 329–357 (2003)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Candès, E.J., Strohmer, T., Voroninski, V.: PhaseLift: exact and stable signal recovery from magnitude measurements via convex programming. Commun. Pure Appl. Math. 66(8), 1241–1274 (2013)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Deza, M.M., Laurent, M.: Geometry of Cuts and Metrics. Algorithms and Combinatorics, vol. 15. Springer, Berlin (1997)zbMATHGoogle Scholar
  15. 15.
    Diestel, R.: Graph Theory, Volume 173 of Graduate Texts in Mathematics, 5th edn. Springer, Berlin (2018)Google Scholar
  16. 16.
    Fujie, T., Kojima, M.: Semidefinite programming relaxation for nonconvex quadratic programs. J. Global Optim. 10(4), 367–380 (1997)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Laurent, M., Varvitsiotis, A.: A new graph parameter related to bounded rank positive semidefinite matrix completions. Math. Program. 145(1–2, Ser. A), 291–325 (2014)MathSciNetzbMATHGoogle Scholar
  18. 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. 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
  20. 20.
    Madani, R., Sojoudi, S., Fazelnia, G., Lavaei, J.: Finding low-rank solutions of sparse linear matrix inequalities using convex optimization. SIAM J. Optim. 27(2), 725–758 (2017)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Pataki, G.: On the rank of extreme matrices in semidefinite programs and the multiplicity of optimal eigenvalues. Math. Oper. Res. 23, 339–358 (1998)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52(3), 471–501 (2010)MathSciNetzbMATHGoogle Scholar
  23. 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. 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
  25. 25.
    Smale, S.: On the average number of steps of the simplex method of linear programming. Math. Program. 27(3), 241–262 (1983)MathSciNetzbMATHGoogle Scholar
  26. 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
  27. 27.
    Sojoudi, S., Lavaei, J.: Exactness of semidefinite relaxations for nonlinear optimization problems with underlying graph structure. SIAM J. Optim. 24(4), 1746–1778 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28(2), 246–267 (2003)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Todd, M.: Semidefinite optimization. Acta Numer. 10, 515–560 (2001)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Todd, M.J.: Polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems. Math. Program. 35(2), 173–192 (1986)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Todd, M.J., Tunçel, L., Ye, Y.: Characterizations, bounds, and probabilistic analysis of two complexity measures for linear programming problems. Math. Program. 90(1, Ser. A), 59–69 (2001)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Yang, B., Anstreicher, K., Burer, S.: Quadratic programs with hollows. Math. Program. 170(2), 541–553 (2018).  https://doi.org/10.1007/s10107-017-1157-0 MathSciNetzbMATHGoogle Scholar
  34. 34.
    Ye, Y.: Toward probabilistic analysis of interior-point algorithms for linear programming. Math. Oper. Res. 19(1), 38–52 (1994)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Ye, Y.: Approximating quadratic programming with bound and quadratic constraints. Math. Program. 81(2), 219–226 (1999)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14(1), 245–267 (2003)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2019

Authors and Affiliations

  1. 1.Department of Management SciencesUniversity of IowaIowa CityUSA
  2. 2.Department of Management Science and Engineering, Institute of Computational and Mathematical EngineeringStanford UniversityStanfordUSA

Personalised recommendations