Skip to main content
SpringerLink
Log in

New global algorithms for quadratic programming with a few negative eigenvalues based on alternative direction method and convex relaxation

  • Full Length Paper
  • Published:
Mathematical Programming Computation Aims and scope Submit manuscript

Abstract

We consider a quadratic program with a few negative eigenvalues (QP-r-NE) subject to linear and convex quadratic constraints that covers many applications and is known to be NP-hard even with one negative eigenvalue (QP1NE). In this paper, we first introduce a new global algorithm (ADMBB), which integrates several simple optimization techniques such as alternative direction method, and branch-and-bound, to find a globally optimal solution to the underlying QP within a pre-specified \(\epsilon \)-tolerance. We establish the convergence of the ADMBB algorithm and estimate its complexity. Second, we develop a global search algorithm (GSA) for QP1NE that can locate an optimal solution to QP1NE within \(\epsilon \)-tolerance and estimate the worst-case complexity bound of the GSA. Preliminary numerical results demonstrate that the ADMBB algorithm can effectively find a global optimal solution to large-scale QP-r-NE instances when \(r\le 10\), and the GSA outperforms the ADMBB for most of the tested QP1NE instances. The software reviewed as part of this submission was given the DOI (digital object identifier) https://doi.org/10.5281/zenodo.1344739.

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

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. It is worth mentioning that in [8], the authors developed a global algorithm for QPs with a small number of negative eigenvalues. However, the algorithm was tested only on small scale instances.

  2. All the quadratic convex optimization problems solved in the GSA algorithm are in form of (36), (37) or (44).

References

  1. Anjos, M .F., Lasserre, J .B.: Handbook of Semidefinite, Conic and Polynomial Optimization: Theory, Algorithms, Software and Applications. International Series in Operational Research and Management Science, p. 166. Springer, Berlin (2001)

    Google Scholar 

  2. Boyd, S., Parikh, N., Chu, E., Peleato, B., Eckstein, J.: Distributed optimization and statistical learning via the alternating direction method of multipliers. Found. Trends Mach. Learn. 3(1), 1–122 (2010)

    Article  MATH  Google Scholar 

  3. Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  4. Burer, S., Vandenbussche, D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Program. 113(2), 259–282 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burer, S., Vandenbussche, D.: Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound. Comput. Optim. Appl. 43(2), 181–195 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cambini, R., Salvi, F.: A branch and reduce approach for solving a class of low rank d.c. programs. J. Comput. Appl. Math. 233, 492–501 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cambini, R., Salvi, F.: Solving a class of low rank d.c. programs via a branch and bound approach: a computational experience. Oper. Res. Lett. 38(5), 354–357 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cambini, R., Sodini, C.: A finite algorithm for a particular d.c. quadratic programming problem. Ann. Oper. Res. 117(1), 33–49 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, J., Burer, S.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Program. Comput. 4, 33–52 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Floudas, C .A., Visweswaran, V.: Quadratic optimization. In: Horst, R., Pardalos, P.M. (eds.) Handbook of Global Optimization, pp. 217–270. Kluwer Academic Publishers, Boston (1994)

    MATH  Google Scholar 

  11. Gorski, J., Pfeuffer, F., Klamroth, K.: Biconvex sets and optimization with biconvex functions: a survey and extensions. Math. Methods Oper. Res. 66, 373–407 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gould, N .I .M., Toint, P .L.: Numerical methods for large-scale non-convex quadratic programming. In: Siddiqi, A.H., Kocvara, M. (eds.) Trends in Industrial and Applied Mathematics. Applied Optimization, vol. 72, pp. 149–179. Springer, Boston (2002)

    Chapter  Google Scholar 

  13. Goyal, V., Genc-Kaya, L., Ravi, R.: An FPTAS for minimizing the product of two non-negative linear cost functions. Math. Program. 126, 401–405 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Horst, R., Thoai, N.V.: DC programming: overview. J. Optim. Theory Appl. 103(1), 1–43 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. IBM ILOG CPLEX. IBM ILOG CPLEX 12.3 User’s Manual for CPLEX, 89 (2011)

  16. Kim, S., Kojima, M.: Exact solutions of some nonconvex quadratic optimization problems via SDP and SOCP relaxations. Comput. Optim. Appl. 26, 143–154 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kojima, M., Tunçel, L.: Cones of matrices and successive convex relaxations of nonconvex sets. SIAM J. Optim. 10, 750–778 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kojima, M., Tunçel, L.: Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization. Math. Program. 89(1), 79–111 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Konno, H.: A cutting plane algorithm for solving bilinear programs. Math. Program. 11, 14–27 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lasserre, J.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Laurent, M.: A comparison of the sherali-adams, lovász-schrijver, and lasserre relaxations for 0–1 programming. Math. Oper. Res. 28(3), 470–496 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Le Thi, H.A., Pham, T.: Dinh. Solving a class of linearly constrained indefinite quadratic problems by D.C. algorithms. J. Glob. Optim. 11, 253–285 (1997)

    Article  MATH  Google Scholar 

  23. Le Thi, H.A., Pham Dinh, T.: A branch and bound method via d.c. optimization algorithms and ellipsoidal technique for box constrained nonconvex quadratic problems. J. Glob. Optim. 13, 171–206 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  24. Le Thi, H.A.: An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints. Math. Program. Ser. A 87(3), 401–426 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Loridan, P.: Necessary conditions for \(\epsilon \)-optimality. Math. Program. Stud. 19, 140–152 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 2(1), 166–190 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. Matsui, T.: NP-hardness of linear multiplicative programming and related problems. J. Glob. Optim. 9(2), 113–119 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  28. Nie, J.: Optimality conditions and finite convergence of lasserre’s hierarchy. Math. Program. 146(1–2), 97–121 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Palacios-Gomez, F., Lasdon, L., Enquist, M.: Nonlinear optimization by successive linear programming. Manag. Sci. 28(10), 1106–1120 (1982)

    Article  MATH  Google Scholar 

  30. Pardalos, P.M., Rodgers, G.: Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45, 131–144 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pardalos, P.M.: Global optimization algorithms for linearly constrained indefinite quadratic problems. Comput. Math. Appl. 21, 87–97 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pardalos, P.M., Schnitger, G.: Checking local optimality in constrained quadratic programming is NP-hard. Oper. Res. Lett. 7(1), 33–35 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Glob. Optim. 1, 15–22 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  34. Peng, J., Zhu, T.: A nonlinear semidefinite programming approch for the worst-case linear optimization under uncertainties. Math. Program. 152(1), 593–614 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pham Dinh, T., Le Thi, H. A.: Recent advances in DC programming and DCA. In: Transactions on Computational Intelligence XIII, pp. 1–37. Springer, Berlin (2014)

  36. Sahinidis, N.V.: BARON: a general purpose global optimization software package. J. Glob. Optim. 8, 201–205 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  37. Saxena, A., Bonami, P., Lee, J.: Convex relaxation of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. Ser. B 124, 383–411 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  38. Saxena, A., Bonami, P., Lee, J.: Convex relaxation of nonconvex mixed integer quadratically constrained programs: projected formulations. Math. Program. Ser. A 130, 359–413 (2011)

    Article  MATH  Google Scholar 

  39. Sherali, H., Adams, W.: A hierarchy of relaxation between the continuous and convex hull representations. SIAM J. Discret. Math. 3, 411–430 (1990)

    Article  MATH  Google Scholar 

  40. Sherali, H.D., Tuncbilek, C.H.: A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Global Optim. 7(1), 1–31 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  41. Vandenbussche, D., Nemhauser, G.: A branch-and-cut algorithm for nonconvex quadratic programming with box constraints. Math. Program. 102, 559–575 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  42. Vandenbussche, D., Nemhauser, G.: A polyhedral study of nonconvex quadratic programs with box constraints. Math. Program. 102(3), 531–557 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  43. Vavasis, S.A.: Approximation algorithms for indefinite quadratic programming. Math. Program. 57, 279–311 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  44. Visweswaran, V., Floudas, C.: New properties and computational improvement of the GOP algorithm for problems with quadratic objective function and constraints. J. Glob. Optim. 3(3), 439–462 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite Programming. Kluwer Academic Publishers, Boston (2000)

    Book  MATH  Google Scholar 

  46. Ye, Y.: On the complexity of approximating a KKT point of quadratic programming. Math. Program. 80, 195–211 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  47. Zhang, S.: Quadratic maximization and semidefinite relaxation. Math. Program. 87, 453–465 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  48. Zhang, Y.J., So, A.M.-C.: Optimal spectrum sharing in MIMO cognitive radio networks via semidefinite programming. IEEE J. Sel. Areas Commun. 29, 362–373 (2011)

    Article  Google Scholar 

Download references

Acknowledgements

We would like to thank all the anonymous reviewers and the associate editor for their useful suggestions that has helped to substantially improve the presentation of this work.

Author information

Authors and Affiliations

  1. Department of Management Science & Engineering, College of Economics and Management, Zhejiang University of Technology, Hangzhou, 310032, Zhejiang, P. R. China

    Hezhi Luo

  2. Department of Applied Mathematics, College of Science, Zhejiang University of Technology, Hangzhou, 310032, Zhejiang, P. R. China

    Xiaodi Bai

  3. Department of Industrial Engineering, University of Houston, Houston, TX, 77204, USA

    Gino Lim & Jiming Peng

Authors
  1. Hezhi Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaodi Bai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gino Lim
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jiming Peng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiming Peng.

Additional information

The research of Hezhi Luo is jointly supported by NSFC Grants 11871433 and 11371324 and the Zhejiang Provincial NSFC Grants LY17A010023 and LY18A010011.

The research of Xiaodi Bai is supported by NSFC Grants 11371103 and 11701511.

The research of Jiming Peng is supported by NSF Grants CMMI-1131690 and CMMI-1537712.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, H., Bai, X., Lim, G. et al. New global algorithms for quadratic programming with a few negative eigenvalues based on alternative direction method and convex relaxation . Math. Prog. Comp. 11, 119–171 (2019). https://doi.org/10.1007/s12532-018-0142-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12532-018-0142-9

Keywords

Mathematics Subject Classification

Search

Navigation