Skip to main content
Springer Nature Link
Log in

A dynamic inequality generation scheme for polynomial programming

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

Hierarchies of semidefinite programs have been used to approximate or even solve polynomial programs. This approach rapidly becomes computationally expensive and is often tractable only for problems of small size. In this paper, we propose a dynamic inequality generation scheme to generate valid polynomial inequalities for general polynomial programs. When used iteratively, this scheme improves the bounds without incurring an exponential growth in the size of the relaxation. As a result, the proposed scheme is in principle scalable to large general polynomial programming problems. When all the variables of the problem are non-negative or when all the variables are binary, the general algorithm is specialized to a more efficient algorithm. In the case of binary polynomial programs, we show special cases for which the proposed scheme converges to the global optimal solution. We also present several examples illustrating the computational behavior of the scheme and provide comparisons with Lasserre’s approach and, for the binary linear case, with the lift-and-project method of Balas, Ceria, and Cornuéjols.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Anjos, M., Lasserre, J. (eds.): Handbook on Semidefinite, Conic and Polynomial Optimization. International Series in Operations Research and Management Science. Springer, Berlin (2012)

  2. Bai, Y., de Klerk, E., Pasechnik, D., Sotirov, R.: Exploiting group symmetry in truss topology optimization. Optim. Eng. 10(3), 331–349 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)

    Article  MATH  Google Scholar 

  4. Bayer, C., Teichmann, J.: The proof of Tchakaloff’s theorem. Proc. Am. Math. Soc. 134, 3035–3040 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. de Klerk, E.: Exploiting special structure in semidefinite programming: a survey of theory and applications. Eur. J. Oper. Res. 201(1), 1–10 (2010)

    Article  MATH  Google Scholar 

  6. de Klerk, E., Pasechnik, D.: Approximation of the stability number of a graph via copositive programming. SIAM J. Optim. 12(4), 875–892 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. de Klerk, E., Pasechnik, D., Schrijver, A.: Reduction of symmetric semidefinite programs using the regular \(^*\)-representation. Math. Program. 109(2–3), 613–624 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. de Klerk, E., Sotirov, R.: Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem. Math. Program. 122(2), 225–246 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Deza, M., Laurent, M.: Applications of cut polyhedra-I. J. Comput. Appl. Math. 55(2), 191–216 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Deza, M., Laurent, M.: Applications of cut polyhedra-II. J. Comput. Appl. Math. 55(2), 217–247 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gatermann, K., Parrilo, P.: Symmetry groups, semidefinite programs, and sums of squares. J. Pure Appl. Algebra 192(1–3), 95–128 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ghaddar, B.: New conic optimization techniques for solving binary polynomial programming problems. PhD thesis, University of Waterloo, 2011. http://uwspace.uwaterloo.ca/bitstream/10012/6139/1/Ghaddar_Bissan.pdf

  13. Ghaddar, B., Vera, J.C., Anjos, M.F.: Second-order cone relaxations for binary quadratic polynomial programs. SIAM J. Optim. 21(1), 391–414 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kim, S., Kojima, M., Toint, P.: Recognizing underlying sparsity in optimization. Math. Program. 9(2), 273–303 (2009)

    Article  MathSciNet  Google Scholar 

  15. Lasserre, J.: An explicit equivalent positive semidefinite program for nonlinear 0–1 programs. SIAM J. Optim. 12(3), 756–769 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Lasserre, J.: Semidefinite programming versus LP relaxations for polynomial programming. Math. Oper. Res. 27(2), 347–360 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Laurent, M.: A comparison of the sherali-adams, lovász-schrijver and lasserre relaxations for 0–1 programming. Math. Oper. Res. 28, 470–496 (2001)

    Article  MathSciNet  Google Scholar 

  19. Laurent, M.: Semidefinite representations for finite varieties. Math. Program. 109(Ser. A), 1–26 (2007)

  20. Laurent, M.: Sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry, volume 149 of The IMA Volumes in Mathematics and its Applications, vol. 149, pp. 157–270. Springer, Berlin (2009)

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Nesterov, Y.: Structure of non-negative polynomials and optimization problems. Technical report, Technical Report 9749, CORE, (1997)

  23. Nie, J., Demmel, J.: Sparse SOS relaxations for minimizing functions that are summation of small polynomials. SIAM J. Optim. 19(4), 1534–1558 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Nie, J., Demmel, J., Sturmfels, B.: Minimizing polynomials via sum of squares over the gradient ideal. Math. Program. Ser. A B 106(3), 587–606 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  25. Parrilo, P.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD thesis, Department of Control and Dynamical Systems, California Institute of Technology, Pasadena, California, (2000)

  26. Parrilo, P.: An explicit construction of distinguished representations of polynomials nonnegative over finite sets. Technical report, IFA Technical Report AUT02-02, Zurich-Switzerland, (2002)

  27. Parrilo, P.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96(2), 293–320 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  28. Parrilo, P., Sturmfels, B.: Minimizing polynomial functions, algorithmic and quantitative real algebraic geometry. DIMACS Ser. Discrete Math. Theor. Comput. Sci. 60, 83–89 (2003)

    MathSciNet  Google Scholar 

  29. Peña, J., Vera, J.C., Zuluaga, L.: Computing the stability number of a graph via linear and semidefinite programming. SIAM J. Optim. 18(1), 87–105 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. Peña, J. F., Vera, J. C., Zuluaga, L. F.: Exploiting equalities in polynomial programming. Oper. Res. Lett. 36(2), 223–228 (2008)

  31. Pisinger, D.: The quadratic knapsack problem-a survey. Discrete App. Math. 155(5), 623–648 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Pólik, I., Terlaky, T.: A survey of the \(\cal {S}\)-lemma. SIAM Rev. 49, 371–418 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Putinar, M.: Positive polynomials on compact semi-algebraic sets. Indiana Univ. Math. J. 42, 969–984 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sherali, H.D., Adams, W.P.: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems. SIAM J. Discrete Math. 3(3), 411–430 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sherali, H.D., Tuncbilek, C.H.: Comparison of two reformulation-linearization technique based linear programming relaxations for polynomial programming problems. J. Glob. Optim. 10(4), 381–390 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  36. Shor, N.: A class of global minimum bounds of polynomial functions. Cybernetics 23(6), 731–734 (1987)

    Article  MATH  Google Scholar 

  37. Tchakaloff, V.: Formules de cubature mécanique à coefficients non négatifs. Bull. Sci. Math. 81, 123–134 (1957)

    MathSciNet  MATH  Google Scholar 

  38. Wolkowicz, H., Saigal, R., Vandenberghe, L.: Handbook of Semidefinite programming-Theory, Algorithms, and Applications. Kluwer, Dordrecht (2000)

  39. Zuluaga, L., Vera, J. C., Peña, J.: LMI approximations for cones of positive semidefinite forms. SIAM J. Optim. 16(4), 1076–1091 (2006)

  40. Zuluaga, L. F.: A conic programming approach to polynomial optimization problems: theory and applications. PhD thesis, The Tepper School of Business, Carnegie Mellon University, Pittsburgh, (2004)

Download references

Acknowledgments

The authors sincerely thank the associate editor and two anonymous referees for their many helpful comments and suggestions.

Author information

Authors and Affiliations

  1. IBM Research, Dublin Technology Campus, Damastown Ind. Park, Mulhuddart, 15, Dublin, Ireland

    Bissan Ghaddar

  2. Tilburg School of Economics and Management, Tilburg University, Tilburg, The Netherlands

    Juan C. Vera

  3. Canada Research Chair in Discrete Nonlinear Optimization in Engineering, GERAD and École Polytechnique de Montréal, Montréal, QC, H3C 3A7, Canada

    Miguel F. Anjos

Authors
  1. Bissan Ghaddar
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Juan C. Vera
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Miguel F. Anjos
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Miguel F. Anjos.

Additional information

An extended abstract version of this paper has appeared in the Proceedings of IPCO 2011.

B. Ghaddar: Research supported by a Canada Graduate Scholarship from the Natural Sciences and Engineering Research Council of Canada.

M. F. Anjos: Research partially supported by the Natural Sciences and Engineering Research Council of Canada, and by a Humboldt Research Fellowship.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ghaddar, B., Vera, J.C. & Anjos, M.F. A dynamic inequality generation scheme for polynomial programming. Math. Program. 156, 21–57 (2016). https://doi.org/10.1007/s10107-015-0870-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-015-0870-9

Keywords

Mathematics Subject Classification