Skip to main content

Exact quadratic convex reformulations of mixed-integer quadratically constrained problems

Abstract

We propose a solution approach for the general problem (QP) of minimizing a quadratic function of bounded integer variables subject to a set of quadratic constraints. The resolution is based on the reformulation of the original problem (QP) into an equivalent quadratic problem whose continuous relaxation is convex, so that it can be effectively solved by a branch-and-bound algorithm based on quadratic convex relaxation. We concentrate our efforts on finding a reformulation such that the continuous relaxation bound of the reformulated problem is as tight as possible. Furthermore, we extend our method to the case of mixed-integer quadratic problems with the following restriction: all quadratic sub-functions of purely continuous variables are already convex. Finally, we illustrate the different results of the article by small examples and we present some computational experiments on pure-integer and mixed-integer instances of (QP). Most of the considered instances with up to 53 variables can be solved by our approach combined with the use of Cplex.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. Adams, W.P., Sherali, H.D.: Reformulation linearization technique for discrete optimization problems. In: Pardalos, Panos M., Du, Ding-Zhu, Graham, Ronald L. (eds.) Handbook of Combinatorial Optimization, pp. 2849–2896. Springer, New York (2013)

    Google Scholar 

  2. Al-Khayyal, F.A., Larsen, C., Van Voorhis, T.: A relaxation method for nonconvex quadratically constrained programs. J. Glob. Optim. 6, 215–230 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  3. Anjos, M.F., Lasserre, J.B.: Handbook of semidefinite, conic and polynomial optimization: theory, algorithms, software and applications. Int. Ser. Oper. Res. Manag. Sci. 166, 1–22 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  4. Audet, C., Hansen, P., Jaumard, B., Savard, G.: A branch and cut algorithm for non-convex quadratically constrained quadratic programming. Math. Program. 87(1), 131–152 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  5. Audet, C., Hansen, P., Savard, G.: Essays and Surveys in Global Optimization. GERAD 25th anniversary series. Springer, New York (2005)

    Book  MATH  Google Scholar 

  6. Bao, X., Sahinidis, N.V., Tawarmalani, M.: Semidefinite relaxations for quadratically constrained quadratic programming: a review and comparisons. Math. Program. 129, 129–157 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  7. Billionnet, A., Elloumi, S., Lambert, A.: Linear reformulations of integer quadratic programs. In: MCO 2008, September 8–10, pp. 43–51 (2008)

  8. Billionnet, A., Elloumi, S., Lambert, A.: Extending the QCR method to the case of general mixed integer program. Math. Program. 131(1), 381–401 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  9. Borchers, B.: Csdp, a c library for semidefinite programming. Optim. Methods Softw. 11(1), 613–623 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  10. Buchheim, C., Wiegele, A.: Semidefinite relaxations for non-convex quadratic mixed-integer programming. Math. Program. 141(1–2), 435–452 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  11. Burer, S., Letchford, A.: Non-convex mixed-integer nonlinear programming: a survey. Surv. Oper. Res. Manag. Sci. 17(2), 97–106 (2012)

    MathSciNet  Google Scholar 

  12. Floudas, C.A.: Deterministic Global Optimization. Kluwer Academic Publishing, Dordrecht (2000)

    Book  MATH  Google Scholar 

  13. Garey, M.R., Johnson, D.S.: Computers and Intractability: A guide to the Theory of NP-Completness. W.H. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  14. IBM-ILOG. Ibm ilog cplex 12.5 reference manual. http://pic.dhe.ibm.com/infocenter/cosinfoc/v12r2/index.jsp (2013)

  15. Lambert, A.: IQCP/MIQCP: Library of integer and mixed-integer quadratic quadratically constrained programs. http://cedric.cnam.fr/~lambe_a1/smiqp/Library/iqcp_miqcp.html (2013)

  16. Liberti, L., Maculan, N.: Global optimization: From Theory to Tmplementation. Chapter: Nonconvex optimization and its applications. Springer, New York (2006)

    Book  MATH  Google Scholar 

  17. Linderoth, J.: A simplicial branch-and-bound algorithm for solving quadratically constrained quadratic programs. Math. Program. 103, 251–282 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  18. McCormick, G.P.: Computability of global solutions to factorable non-convex programs: Part i: convex underestimating problems. Math. Program. 10(1), 147–175 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  19. Misener, R., Floudas, C.A.: Global optimization of mixed-integer quadratically-constrained quadratic programs (MIQCQP) through piecewise-linear and edge-concave relaxations. Math. Program. 136, 155–182 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  20. Misener, R., Floudas, C.A.: GloMIQO: global mixed-integer quadratic optimizer. J. Glob. Optim. 57(1), 3–50 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  21. Raber, U.: A simplicial branch-and-bound method for solving nonconvex all-quadratic programs. J. Glob. Optim. 13, 417–432 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  22. Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Math. Program. 130, 359–413 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  23. Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-integer Nonlinear Programming. Kluwer Academic Publishing, Dordrecht (2002)

    Book  MATH  Google Scholar 

  24. Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103(2), 225–249 (2005)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Amélie Lambert.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Billionnet, A., Elloumi, S. & Lambert, A. Exact quadratic convex reformulations of mixed-integer quadratically constrained problems. Math. Program. 158, 235–266 (2016). https://doi.org/10.1007/s10107-015-0921-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-015-0921-2

Keywords

  • Integer quadratic programming
  • Equivalent convex reformulation
  • Semidefinite programming
  • Branch-and-bound algorithm

Mathematics Subject Classification

  • 90C11
  • 90C20