Skip to main content
SpringerLink
Log in

KKT-based primal-dual exactness conditions for the Shor relaxation

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this work we present some exactness conditions for the Shor relaxation of diagonal (or, more generally, diagonalizable) QCQPs, which extend the conditions introduced in different recent papers about the same topic. It is shown that the Shor relaxation is equivalent to two convex quadratic relaxations. Then, sufficient conditions for the exactness of the relaxations are derived from their KKT systems. It will be shown that, in some cases, by this derivation previous conditions in the literature, which can be viewed as dual conditions, since they only involve the Lagrange multipliers appearing in the KKT systems, can be extended to primal-dual conditions, which also involve the primal variables appearing in the KKT systems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

There are no data that support the findings of this study.

References

  1. Ai, W., Zhang, S.: Strong duality for the CDT subproblem: a necessary and sufficient condition. SIAM J. Optim. 19(4), 1735–1756 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beck, A., Pan, D.: A branch and bound algorithm for nonconvex quadratic optimization with ball and linear constraint. J. Global Optim. 69, 309–342 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ben-Tal, A., den Hertog, D.: Hidden conic quadratic representation of some nonconvex quadratic optimization problems. Math. Program. 143, 1–29 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bienstock, D., Michalka, A.: Polynomial solvability of variants of the trust-region subproblem. In: SODA ’14 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, 380–390 (2014)

  5. Bienstock, D.: A note on polynomial solvability of the CDT problem. SIAM J. Optim. 26, 488–498 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burer, S., Anstreicher, K.M.: Second-order-cone constraints for extended trust-region subproblems. SIAM J. Optim. 23(1), 432–451 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burer, S., Yang, B.: The Trust Region Subproblem with non-intersecting linear constraints. Math. Program. 149(1), 253–264 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  8. Burer, S., Ye, Y.: Exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs. Math. Program. 181, 1–17 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  9. Burer, S., Ye, Y.: Erratum to: exact semidefinite formulations for a class of (random and non-random) nonconvex quadratic programs. Math. Program. 190(1), 845–848 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  10. Consolini, L., Locatelli, M.: On the complexity of quadratic programming with two quadratic constraints. Math. Program. 164, 91–128 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fradkov, A.L., Yakubovich, V.A.: The S-procedure and duality relations in nonconvex problems of quadratic programming. Vestnik Leningrad Univ. Math. 6, 101–109 (1979)

    MATH  Google Scholar 

  12. Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion I: general framework. SIAM J. Optim. 11(3), 647–674 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jeyakumar, V., Li, G.: Trust-region problems with linear inequality constraints: exact SDP relaxation, global optimality and robust optimization. Math. Program. 147(1), 171–206 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Jeyakumar, V., Li, G.: Exact second-order cone programming relaxations for some nonconvex minimax quadratic optimization problems. SIAM J. Optim. 28(1), 760–787 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Locatelli, M.: Exactness conditions for an SDP relaxation of the extended trust region problem. Optim. Lett. 10, 1141–1151 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sakaue, S., Nakatsukasa, Y., Takeda, A., Iwata, S.: Solving generalized CDT problems via two-parameter eigenvalues. SIAM J. Optim. 26, 1669–1694 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Shor, N.: Quadratic optimization problems. Soviet J. Comput. Syst. Sci. 25, 1–11 (1987)

    MathSciNet  MATH  Google Scholar 

  18. Sojoudi, S., Lavaei, J.: Exactness of semidefinite relaxations for nonlinear optimization problems with underlying graph structure. SIAM J. Optim. 24(4), 1746–1778 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sturm, J.F., Zhang, S.: On cones of nonnegative quadratic functions. Math. Oper. Res. 28(2), 246–267 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Yang, B., Burer, S.: A two-varable approach to the two-trust region subproblem. SIAM J. Optim. 26(1), 661–680 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ye, Y., Zhang, S.: New results on quadratic minimization. SIAM J. Optim. 14, 245–267 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Yuan, J., Wang, M., Ai, W., Shuai, T.: New results on narrowing the duality gap on the extended Celis-Dennis-Tapia problem. SIAM J. Optim. 27(2), 890–909 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, A.L., Kilinç-Karzan, F.: On the tightness of SDP relaxations of QCQPs. Math. Program. 193(1), 33–73 (2022)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria e Architettura, Università degli Studi di Parma, Parma, Italy

    M. Locatelli

Authors
  1. M. Locatelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. Locatelli.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Locatelli, M. KKT-based primal-dual exactness conditions for the Shor relaxation. J Glob Optim 86, 285–301 (2023). https://doi.org/10.1007/s10898-022-01258-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-022-01258-5

Keywords

Search

Navigation