Skip to main content
SpringerLink
Log in

On zero duality gap in nonconvex quadratic programming problems

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

Abstract

We present in this paper new sufficient conditions for verifying zero duality gap in nonconvex quadratically/linearly constrained quadratic programs (QP). Based on saddle point condition and conic duality theorem, we first derive a sufficient condition for the zero duality gap between a quadratically constrained QP and its Lagrangian dual or SDP relaxation. We then use a distance measure to characterize the duality gap for nonconvex QP with linear constraints. We show that this distance can be computed via cell enumeration technique in discrete geometry. Finally, we revisit two sufficient optimality conditions in the literature for two classes of nonconvex QPs and show that these conditions actually imply zero duality gap.

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

References

  1. Avis D., Fukuda K.: Reverse search for enumeration. Discrete Appl. Math. 65, 21–46 (1996)

    Article  Google Scholar 

  2. Beck A., Eldar Y.C.: Strong duality in nonconvex quadratic optimization with two quadratic constraints. SIAM J. Optim. 17, 844–860 (2006)

    Article  Google Scholar 

  3. Beck A., Teboulle M.: Global optimality conditions for quadratic optimization problems with binary constraints. SIAM J. Optim. 11, 179–188 (2000)

    Article  Google Scholar 

  4. Ben-Tal, A.: Conic and robust optimization. Lecture notes. Universita di Roma La Sapienzia, Rome, Italy (2002). Available at: http://ie.technion.ac.il/Home/Users/morbt/rom.pdf

  5. Ben-Tal A., Teboulle M.: Hidden convexity in some nonconvex quadratically constrained quadratic programming. Math. Program. 72, 51–63 (1996)

    Google Scholar 

  6. Carraresi P., Farinaccio F., Malucelli F.: Testing optimality for quadratic 0-1 problems. Math. Program. 85, 407–421 (1999)

    Article  Google Scholar 

  7. Chaovalitwongse W., Androulakis I.P., Pardalos P.M.: Quadratic integer programming: complexity and equivalent forms. In: Floudas, C.A., Pardalos, P.M (eds) Encyclopedia of Optimization, pp. 3153–3159. Springer, New York (2009)

    Chapter  Google Scholar 

  8. Chaovalitwongse W., Pardalos P.M., Prokopyev O.A.: A new linearization technique for multi-quadratic 0-1 programming problems. Oper. Res. Lett. 32, 517–522 (2004)

    Article  Google Scholar 

  9. De Angelis P.L., Toraldo G.: Quadratic programming with bound constraints. In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization, pp. 3161–3166. Springer, New York (2009)

    Chapter  Google Scholar 

  10. Edelsbrunner H.: Algorithms in Combinatorial Geometry. Springer, Heidelberg (1987)

    Google Scholar 

  11. Fang S.C., Gao D.Y., Sheu R.L., Wu S.Y.: Canonical dual approach to solve 0-1 quadratic programming problems. J. Ind. Manage. Optim. 4, 125–142 (2008)

    Article  Google Scholar 

  12. Fradkov A.L., Yakubovich V.A.: The S-procedure and the duality relation in convex quadratic programming problems. Vestnik Leningrad Univ. 1, 81–87 (1973)

    Google Scholar 

  13. Fujie T., Kojima M.: Semidefinite programming relaxation for nonconvex quadratic programs. J. Glob. Optim. 10, 367–380 (1997)

    Article  Google Scholar 

  14. Gao D.Y., Ruan N., Sherali H.D.: Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality. J. Glob. Optim. 45, 473–497 (2009)

    Article  Google Scholar 

  15. Hiriart-Urruty J.B.: Conditions for global optimality 2. J. Glob. Optim. 13, 349–367 (1998)

    Article  Google Scholar 

  16. Hiriart-Urruty J.B.: Global optimality conditions in maximizing a convex quadratic function under convex quadratic constraints. J. Glob. Optim. 21, 443–453 (2001)

    Article  Google Scholar 

  17. Jeyakumar V., Rubinov A.M., Wu Z.Y.: Sufficient global optimality conditions for non-convex quadratic minimization problems with box constraints. J. Glob. Optim. 36, 471–481 (2006)

    Article  Google Scholar 

  18. Jeyakumar V., Rubinov A.M., Wu Z.Y.: Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Math. Program. 110, 521–541 (2007)

    Article  Google Scholar 

  19. 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  Google Scholar 

  20. More J.J.: Generalizations of the trust region problem. Optim. Methods Softw. 2, 189–209 (1993)

    Article  Google Scholar 

  21. Nesterov, Y., Nemirovski, A.: Interior-point polynomial algorithms in convex programming. Society for Industrial & Applied (1994)

  22. Pardalos P.M., Chaovalitwongse W., Iasemidis L.D., Chris Sackellares J., Shiau D.S., Carney P.R., Prokopyev O.A., Yatsenko V.A.: Seizure warning algorithm based on optimization and nonlinear dynamics. Math. Program. 101, 365–385 (2004)

    Article  Google Scholar 

  23. Pardalos P.M., Glick J.H., Rosen J.B.: Global minimization of indefinite quadratic problems. Computing 39, 281–291 (1987)

    Article  Google Scholar 

  24. Pardalos, P.M., Resende, M. (eds): Handbook of Applied Optimization. Oxford University Press, New York (2002)

    Google Scholar 

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

    Article  Google Scholar 

  26. Pinar M.Ç.: Sufficient global optimality conditions for bivalent quadratic optimization. J. Optim. Theory Appl. 122, 433–440 (2004)

    Article  Google Scholar 

  27. Pitsoulis L.: Quadratic programming with bound constraints. In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization, pp. 3170–3171. Springer, New York (2009)

    Chapter  Google Scholar 

  28. Polik I., Terlaky T.: A survey of the S-lemma. SIAM Rev. 49, 371–418 (2007)

    Article  Google Scholar 

  29. Rosen J.B., Pardalos P.M.: Global minimization of large-scale constrained concave quadratic problems by separable programming. Math. Program. 34, 163–174 (1986)

    Article  Google Scholar 

  30. Ruiz J.P., Grossmann I.E.: Exploiting vector space properties to strengthen the relaxation of bilinear programs arising in the global optimization of process networks. Optim. Lett. 5, 1–11 (2011)

    Article  Google Scholar 

  31. Sleumer N.: Output-sensitive cell enumeration in hyperplane arrangements. Nordic J. Comput. 6, 137–161 (1999)

    Google Scholar 

  32. Stern R.J., Wolkowicz H.: Indefinite trust region subproblems and nonsymmetric eigenvalue perturbations. SIAM J. Optim. 5, 286–313 (1995)

    Article  Google Scholar 

  33. Vandenberghe L., Boyd S.: Semidefinite programming. SIAM Rev. 38, 49–95 (1996)

    Article  Google Scholar 

  34. Wang Z.B., Fang S.C., Gao D.Y., Xing W.X.: Global extremal conditions for multi-integer quadratic programming. J. Ind. Manage. Optim. 4, 213–225 (2008)

    Article  Google Scholar 

  35. Wu Z.Y., Bai F.: Global optimality conditions for mixed nonconvex quadratic programs. Optimization 58, 39–47 (2009)

    Article  Google Scholar 

  36. Wu Z.Y., Jeyakumar V., Rubinov A.M.: Sufficient conditions for global optimality of bivalent nonconvex quadratic programs with inequality constraints. J. Optim. Theory Appl. 133, 123–130 (2007)

    Article  Google Scholar 

  37. Yajima Y.: Quadratic knapsack. In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization, pp. 3159–3161. Springer, New York (2009)

    Chapter  Google Scholar 

  38. Ye Y.: Quadratic programming over an ellipsoid. In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization, pp. 3166–3170. Springer, New York (2009)

    Chapter  Google Scholar 

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

    Article  Google Scholar 

  40. Zaslavsky T.: Facing up to arrangements: face-count formulas for partitions of space by hyperplanes. Mem. Am. Math. Soc. 1, 1–101 (1975)

    Google Scholar 

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

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Management Science, School of Management, Fudan University, Shanghai, 200433, People’s Republic of China

    X. J. Zheng, X. L. Sun & Y. F. Xu

  2. Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong

    D. Li

Authors
  1. X. J. Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. X. L. Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Y. F. Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to X. L. Sun.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zheng, X.J., Sun, X.L., Li, D. et al. On zero duality gap in nonconvex quadratic programming problems. J Glob Optim 52, 229–242 (2012). https://doi.org/10.1007/s10898-011-9660-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-011-9660-y

Keywords

Search

Navigation