Computational Optimization and Applications

, Volume 66, Issue 1, pp 1–37 | Cite as

On the solution of nonconvex cardinality Boolean quadratic programming problems: a computational study

  • Ricardo M. Lima
  • Ignacio E. Grossmann


This paper addresses the solution of a cardinality Boolean quadratic programming problem using three different approaches. The first transforms the original problem into six mixed-integer linear programming (MILP) formulations. The second approach takes one of the MILP formulations and relies on the specific features of an MILP solver, namely using starting incumbents, polishing, and callbacks. The last involves the direct solution of the original problem by solvers that can accomodate the nonlinear combinatorial problem. Particular emphasis is placed on the definition of the MILP reformulations and their comparison with the other approaches. The results indicate that the data of the problem has a strong influence on the performance of the different approaches, and that there are clear-cut approaches that are better for some instances of the data. A detailed analysis of the results is made to identify the most effective approaches for specific instances of the data.


Integer programming Quadratic programming Computing science 



The first author acknowledges the support of the Center for Uncertainty Quantification in Computational Science & Engineering. This research used resources in the King Abdullah University of Science and Technology Scientific Computing Center, supported by the Information Technology, Research Computing Team. The authors would like to thank the Center for Advanced Process Decision-making at Carnegie Mellon University for their financial support. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007- 2013) under grant agreement n. PCOFUND-GA-2009-246542 and from the Fundação para a Ciência e Tecnologia, Portugal, under Grant agreement n. DFRH/WIIA/67/2011.

Supplementary material

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  1. 1.
    Krislock, N., Malick, J., Roupin, F.: Improved semidefinite bounding procedure for solving max-cut problems to optimality. Math. Program. Ser. A 143, 61–86 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Wolsey, L.A.: Integer Programming. Wiley Series in Discrete Mathematics and Optimization. Wiley, Chichester (1998)Google Scholar
  3. 3.
    Loiola, E.M., Abreu, N.M.M., Boaventura-Netto, P.O., Hahn, P., Querido, T.: A survey for the quadratic assignment problem. Eur. J. Oper. Res. 176(2), 657–690 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Billionnet, A., Costa, M.C., Sutter, A.: An efficient algorithm for a task allocation problem. J. ACM 39(3), 502–518 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Phillips, A.T., Rosen, J.B.: A quadratic assignment formulation of the molecular conformation problem. J. Glob. Optim. 4(2), 229–241 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Barahona, F., Junger, M., Reinelt, G.: Experiments in quadratic 0–1 programming. Math. Program. 44(2), 127–137 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Caprara, A.: Constrained 0–1 quadratic programming: basic approaches and extensions. Eur. J. Oper. Res. 187(3), 1494–1503 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Padberg, M.: The boolean quadric polytope—some characteristics, facets and relatives. Math. Program. 45(1), 139–172 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Billionnet, A.: Different formulations for solving the heaviest k-subgraph problem. INFOR 43(3), 171–186 (2005)MathSciNetGoogle Scholar
  10. 10.
    Pisinger, D.: Upper bounds and exact algorithms for p-dispersion problems. Comput. Oper. Res. 33(5), 1380–1398 (2006)CrossRefzbMATHGoogle Scholar
  11. 11.
    Marti, Rafael, Gallego, Micael, Duarte, Abraham: A branch and bound algorithm for the maximum diversity problem. Eur. J. Oper. Res. 200(1), 36–44 (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Malick, J., Roupin, F.: Solving k-cluster problems to optimality with semidefinite programming. Math. Program. 136(2, SI), 279–300 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bertsimas, D., Shioda, R.: Algorithm for cardinality-constrained quadratic optimization. Comput. Optim. Appl. 43(1), 1–22 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    IBM ILOG CPLEX Optimization Studio - CPLEX Users Manual - Version 12 Release 6. IBM (2013)Google Scholar
  15. 15.
    Bruglieri, M., Ehrgott, M., Hamacher, H.W., Maffioli, F.: An annotated bibliography of combinatorial optimization problems with fixed cardinality constraints. Discret. Appl. Math. 154(9), 1344–1357 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Porn, R., Nissfolk, O., Jansson, F., Westerlund, T.: The Coulomb glass—modeling and computational experience with a large scale 0-1 QP problem. In: Pistikopoulos, E.N., Georgiadis, M.C., Kokossis, A.C. (eds.) 21st European Symposium on Computer Aided Process Engineering, vol. 29, pp. 658–662 (2011)Google Scholar
  17. 17.
    GUROBI Documentation. GUROBI (2016). Accessed 29 May 2016
  18. 18.
    FICO Xpress Optimization Suite—Xpress-Optimizer—Reference manual. FICO (2009). Accessed 29 May 2016
  19. 19.
    Achterberg, T.: SCIP solving constraint integer programs. Math. Program. 1(1), 1–41 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Androulakis, I.P., Maranas, C.D., Floudas, C.A.: \(\alpha \)BB: A global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7(4), 337–363 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Inc. Lindo systems. LINDOGlobal (2007). Accessed 30 May 2016
  22. 22.
    Belotti, P., Lee, J., Liberti, L., Margot, F., Waechter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Method. Softw. 24(4–5), 597–634 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tawarmalani, M., Sahinidis, N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Misener, Ruth, Floudas, Christodoulos A.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59(2–3), 503–526 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Glover, F., Woolsey, E.: Converting 0–1 polynomial programming problem to a 0–1 linear program. Oper. Res. 2(1), 180–182 (1974)CrossRefzbMATHGoogle Scholar
  26. 26.
    Raman, R., Grossmann, I.E.: Relation between MILP modeling and logical inference for chemical process synthesis. Comput. Chem. Eng. 15(2), 73–84 (1991)CrossRefGoogle Scholar
  27. 27.
    Sherali, H.D., Adams, W.P.: A reformulation-linearization technique for solving discrete and continuous nonconvex problems. Springer US (1999)Google Scholar
  28. 28.
    Billionnet, A., Elloumi, S.: Using a mixed integer quadratic programming solver for the unconstrained quadratic 0–1 problem. Math. Program. 109(1), 55–68 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Pisinger, D.: The quadratic knapsack problem—a survey. Discret. Appl. Math. 155(5), 623–648 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mehrotra, A.: Cardinality constrained Boolean quadratic polytope. Discret. Appl. Math. 79(1–3), 137–154 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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
  32. 32.
    Pardalos, P.M., Rodgers, G.P.: Computational aspects of a branch and bound algorithm for quadratic zero-one programming. Computing 45(2), 131–144 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Sherali, H.D., Lee, Y.H., Adams, W.P.: A simultaneous lifting strategy for identifying new classes of facets for the boolean quadric polytope. Oper. Res. Lett. 17(1), 19–26 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Gueye, S., Michelon, P.: “Miniaturized” linearizations for quadratic 0/1 problems. Ann. Oper. Res. 140(1), 235–261 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Gueye, S., Michelon, P.: A linearization framework for unconstrained quadratic (0–1) problems. Discret. Appl. Math. 157(6), 1255–1266 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Hansen, P., Meyer, C.: Improved compact linearizations for the unconstrained quadratic 0–1 minimization problem. Discrete Applied Mathematics 157, 1267–1290 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Liberti, L.: Compact linearization for binary quadratic problems. 4OR 5(3), 231–245 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Burer, S., Letchford, A.N.: On nonconvex quadratic programming with box constraints. SIAM J. Optim. 20(2), 1073–1089 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Johnson, E.L., Mehrotra, A., Nemhauser, G.L.: Min-cut clustering. Math. Program. 62(1), 133–151 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Faye, A., Trinh, Q.: Polyhedral results for a constrained quadratic 0-1 problem. Technical Report CEDRIC-03-511, CEDRIC laboratory, CNAM-Paris, France (2003)Google Scholar
  41. 41.
    Faye, A., Trinh, Q.A.: A polyhedral approach for a constrained quadratic 0–1 problem. Discret. Appl. Math. 149(1–3), 87–100 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Macambira, E.M., de Souza, C.C.: The edge-weighted clique problem: valid inequalities, facets and polyhedral computations. Eur. J. Oper. Res. 123(2), 346–371 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Glover, F.: Improved linear integer programming formulations of nonlinear integer problems. Manag. Sci. 22, 455–460 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Rothberg, E.: An evolutionary algorithm for polishing mixed integer programming solutions. INFORMS J. Comput. 19(4), 534–541 (2007)CrossRefzbMATHGoogle Scholar
  45. 45.
    Brooke, A., Kendrick, D., Meeraus, A., Raman, R.: GAMS—a user’s guide (1998)Google Scholar
  46. 46.
    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. Hosei University, Tokyo, Japan (2014)Google Scholar
  47. 47.
    Billionnet, A., Elloumi, S., Plateau, M.C.: Improving the performance of standard solvers for quadratic 0–1 programs by a tight convex reformulation: The QCR method. Discret. Appl. Math. 157(6), 1185–1197 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Dinh, T.P., Canh, N.N., Le, T., An, H.: An efficient combined DCA and B&B using DC/SDP relaxation for globally solving binary quadratic programs. J. Glob. Optim. 48(4), 595–632 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Rendl, F., Rinaldi, G., Wiegele, A.: Solving max-cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. Ser. A 121, 307–335 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Lima, R.M., Grossmann, I. E.: Chemical engineering greetings to Prof. Sauro Pierucci, chapter. Computational advances in solving Mixed Integer Linear Programming problems, pp. 151–160, AIDIC (2011)Google Scholar
  51. 51.
    Bixby, R., Rothberg, E.: Progress in computational mixed integer programming—a look back from the other side of the tipping point. Ann. Oper. Res. 49(1), 37–41 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Dolan, E.D., Moré, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201–213 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Bussieck, M.R., Dirkse, S.P., Vigerske, S.: Paver 2.0: an open source environment for automated performance analysis of benchmarking data. J. Glob. Optim. 59(2–3), 259–275 (2014)CrossRefzbMATHGoogle Scholar
  54. 54.
    Lima, R.M.: IBM ILOG CPLEX What is inside of the box? (2010). Accessed 22 Oct 2015

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Computer, Electrical and Mathematical Sciences & Engineering DivisionKing Abdullah University of Science and Technology (KAUST)ThuwalKingdom of Saudi Arabia
  2. 2.Department of Chemical EngineeringCarnegie Mellon UniversityPittsburghUSA

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