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Mathematical Programming Computation

, Volume 10, Issue 3, pp 333–382 | Cite as

Globally solving nonconvex quadratic programming problems with box constraints via integer programming methods

  • Pierre Bonami
  • Oktay Günlük
  • Jeff Linderoth
Full Length Paper
  • 257 Downloads

Abstract

We present effective linear programming based computational techniques for solving nonconvex quadratic programs with box constraints (BoxQP). We first observe that known cutting planes obtained from the Boolean Quadric Polytope (BQP) are computationally effective at reducing the optimality gap of BoxQP. We next show that the Chvátal–Gomory closure of the BQP is given by the odd-cycle inequalities even when the underlying graph is not complete. By using these cutting planes in a spatial branch-and-cut framework, together with a common integrality-based preprocessing technique and a particular convex quadratic relaxation, we develop a solver that can effectively solve a well-known family of test instances. Our linear programming based solver is competitive with SDP-based state of the art solvers on small instances and sparse instances. Most of our computational techniques have been implemented in the recent version of CPLEX and have led to significant performance improvements on nonconvex quadratic programs with linear constraints.

Keywords

Nonconvex quadratic programming Global optimization Boolean Quadric Polytope 

Mathematics Subject Classification

90C20 90C26 90C57 

References

  1. 1.
    An, L.T.H., Tao, P.D.: 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Andersen, M., Dahl, J., Vandenberghe, L.: CVXOPT user’s guide, release 1.1.8 (2015)Google Scholar
  3. 3.
    Anstreicher, K.: On convex relaxations for quadratically constrained quadratic programming. Math. Program. 136, 233–251 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Anstreicher, K., Burer, S.: Computable representations for convex hulls of low-dimensional quadratic forms. Math. Program. 124, 33–43 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Anstreicher, K.M.: Semidefinite programming versus the reformulation-linearization technique for nonconvex quadratically constrained quadratic programming. J. Glob. Optim. 43(2), 471–484 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Barahona, F.: On cuts and matchings in planar graphs. Math. Program. 60, 53,58 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Barahona, F., Grötschel, M., Jünger, M., Reinelt, G.: Experiments in quadratic 01 programming. Math. Program. 44, 127–137 (1989)CrossRefzbMATHGoogle Scholar
  8. 8.
    Barahona, F., Mahjoub, A.: On the cut polytope. Math. Program. 36, 157–173 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bliek, C., Bonami, P., Lodi, A.: Solving mixed-integer quadratic programming problems with IBM-CPLEX: a progress report. In: Proceedings of the Twenty-Sixth RAMP Symposium, pp. 171–180 (2014)Google Scholar
  10. 10.
    Boros, E., Crama, Y., Hammer, P.L.: Chvátal cuts and odd cycle inequalities in quadratic 0–1 optimization. SIAM J. Discrete Math. 5(2), 163–177 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Boros, E., Hammer, P.L.: Cut-polytopes, Boolean quadric polytopes and nonnegative quadratic pseudo-Boolean functions. Math. Oper. Res. 18(1), 245–253 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Burer, S., Monteiro, D.R.: A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. 95(2), 329–357 (2003).  https://doi.org/10.1007/s10107-002-0352-8 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Burer, S.: Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Math. Program. Comput. 2(1), 119 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Burer, S., Chen, J.: Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Program. Comput. 4(1), 33–52 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Burer, S., Letchford, A.: On nonconvex quadratic programming with box constriants. SIAM J. Optim. 20(2), 1073–1089 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Burer, S., Monteiro, R., Choi, C.: SDPLR 1.03-beta user’s guide (short version) (2009). http://sburer.github.io/files/SDPLR-1.03-beta-usrguide.pdf
  17. 17.
    Burer, S., Vandenbussche, D.: Globally solving box-constrained nonconvex quadratic programs with semdefinite-based finite branch-and-bound. Comput. Optim. Appl. 43, 181–195 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Caprara, A., Fischetti, M.: \({\{0, \frac{1}{2}\}}\) chvátal-gomory cuts. Math. Program. 74, 221–235 (1996)zbMATHGoogle Scholar
  19. 19.
    Chvátal, V.: Edmonds polytopes and weakly Hamiltonian graphs. Math. Program. 5, 29–40 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dolan, E., Moré, J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Dong, H.: Relaxing nonconvex quadratic functions by multiple adaptive diagonal perturbations. SIAM J Optim 26(3), 1962–1985 (2014).  https://doi.org/10.1137/140960657 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dong, H., Linderoth, J.: On valid inequalities for quadratic programming with continuous variables and binary indicators. In: IPCO 2013: The Sixteenth Conference on Integer Programming and Combinatorial Optimization, vol. 7801, pp. 169–180. Springer (2013)Google Scholar
  23. 23.
    Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Mon. 64, 275–278 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Hansen, P., Jaumard, B., Ruiz, M., Xiong, J.: Global minimization of indefinite quadratic functions subject to box constraints. Naval Res. Logist. 40(3), 373–392 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Horst, H., Pardalos, P.M., Thoai, V.: Introduction to Global Optimization, 2nd edn. Kluwer, Dordrecht (2000)CrossRefzbMATHGoogle Scholar
  26. 26.
    Koster, A., Zymolka, A., Kutschka, M.: Algorithms to separate 0,1/2-Chvátal–Gomory cuts. Algorithmica 55(2), 375–391 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I—convex underestimating problems. Math. Program. 10, 147–175 (1976)CrossRefzbMATHGoogle Scholar
  28. 28.
    Misener, R., Smadbeck, J.B., Floudas, C.A.: Dynamically-generated cutting planes for mixed-integer quadratically-constrained quadratic programs and their incorporation into GloMIQO 2.0. Optim. Methods Softw. 30, 215–249 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Padberg, M.: The boolean quadric polytope: some characterics, facets, and relatives. Math. Program. 45, 139–172 (1989)CrossRefzbMATHGoogle Scholar
  30. 30.
    Padberg, M.W.: Total unimodularity and the Euler-subgraph problem. Oper. Res. Lett. 7(4), 173–179 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Math. Program. 130, 359–413 (2011). Version with appendix available at http://www.optimization-online.org/DB_FILE/2008/11/2145.pdf
  32. 32.
    Sherali, H., Tuncbilek, C.: A new reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Glob. Optim. 7, 1–31 (1995)CrossRefzbMATHGoogle Scholar
  33. 33.
    Sherali HD, Alameddine AR (1990) An explicit characterization of the convex envelope of a bivariate function over special polytopes. Ann. Oper. Res. Comput. Methods Glob. Optim. 25(1): 197–210Google Scholar
  34. 34.
    Shor, N.Z.: Quadratic optimization problems. Sov. J. Circuits Syst. Sci. 25(6), 1–11 (1987)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Simone, C.D.: The cut polytope and the boolean quadric polytope. Discrete Math. 79, 71–75 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Sturm, J.F.: Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    The MOSEK command line tool. Version 7.1 (revision 51) (2016). http://docs.mosek.com/7.1/tools/index.html
  38. 38.
    Tawarmalani, M., Sahinidis, N.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed integer nonlinear programs: a theoretical and computational study. Math. Program. 99, 563–591 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Vandenbussche, D., Nemhauser, G.L.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102, 559–575 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Yajima, Y., Fujie, T.: A polyhedral approach for nonconvex quadratic programming problems with box constraints. J. Glob. Optim. 13, 151–170 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and The Mathematical Programming Society 2018

Authors and Affiliations

  1. 1.IBM SpainMadridSpain
  2. 2.IBM ResearchYorktown HeightsUSA
  3. 3.Department of Industrial and Systems Engineering, Wisconsin Institutes of DiscoveryUniversity of Wisconsin-MadisonMadisonUSA

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