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Extending the QCR method to general mixed-integer programs

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Abstract

Let (MQP) be a general mixed integer quadratic program that consists of minimizing a quadratic function subject to linear constraints. In this paper, we present a convex reformulation of (MQP), i.e. we reformulate (MQP) into an equivalent program, with a convex objective function. Such a reformulation can be solved by a standard solver that uses a branch and bound algorithm. We prove that our reformulation is the best one within a convex reformulation scheme, from the continuous relaxation point of view. This reformulation, that we call MIQCR (Mixed Integer Quadratic Convex Reformulation), is based on the solution of an SDP relaxation of (MQP). Computational experiences are carried out with instances of (MQP) including one equality constraint or one inequality constraint. The results show that most of the considered instances with up to 40 variables can be solved in 1 h of CPU time by a standard solver.

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References

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

    Book  MATH  Google Scholar 

  2. Billionnet A., Elloumi S.: Using a mixed integer quadratic programming solver for the unconstrained quadratic 0–1 problem. Math. Program. 109, 55–68 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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. Discrete Appl. Math. 157(6), 1185–1197 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bonami P., Biegler L., Conn A., Cornuéjols G., Grossmann I., Laird C., Lee J., Lodi A., Margot F., Sawaya N., Waechter A.: An algorithmic framework for convex mixed integer nonlinear programming. Discrete Optim. 5, 186–204 (2005)

    Article  Google Scholar 

  5. Borchers B.: CSDP, A C library for semidefinite programming. Optim. Methods Softw. 11(1), 613–623 (1999)

    Article  MathSciNet  Google Scholar 

  6. Caprara A.: Constrained 0–1 quadratic programming: basic approaches and extensions. Eur. J. Oper. Res. 187(3), 1494–1503 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cui Y.: Dynamic programming algorithms for the optimal cutting of equal rectangles. Appl.Math. Model. 29, 1040–1053 (2005)

    Article  MATH  Google Scholar 

  8. Floudas C.A.: Deterministic Global Optimization. Kluwer, Dordrecht (2000)

    Google Scholar 

  9. Faye A., Roupin F.: Partial lagrangian relaxation for general quadratic programming, 4’ OR. Q. J. Oper. Res. 5(1), 75–88 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Frangioni A., Gentile C.: Perspective cuts for a class of convex 0–1 mixed integer programs. Math. Program. 106, 225–236 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fu H.L., Shiue L., Cheng X., Du D.Z., Kim J.M.: Quadratic integer programming with application in the chaotic mappings of complete multipartite graphs. J. Optim. Theory Appl. 110(3), 545–556 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  13. Glover F., Woolsey R.E.: Converting the 0–1 polynomial programming problem to a 0–1 linear program. Oper. Res. 22, 180–182 (1974)

    Article  MATH  Google Scholar 

  14. Glover F.: Improved linear integer programming formulations of nonlinear integer problems. Manag. Sci. 22, 455–460 (1975)

    Article  MathSciNet  Google Scholar 

  15. Hammer P.L., Rubin A.A.: Some remarks on quadratic programming with 0–1 variables. Revue Française d’Informatique et de Recherche Opérationnelle. 4(3), 67–79 (1970)

    MathSciNet  Google Scholar 

  16. Helmberg C., Rendl F.: Solving quadratic (0,1)-problems by semidefinite programs and cutting planes. Math. Program. 82, 291–315 (1998)

    MathSciNet  MATH  Google Scholar 

  17. Hua Z.S., Banerjee P.: Aggregate line capacity design for PWB assembly systems. Int. J. Prod. Res. 38(11), 2417–2441 (2000)

    Article  MATH  Google Scholar 

  18. ILOG. ILOG CPLEX 11.0 Reference Manual. ILOG CPLEX Division, Gentilly (2008)

  19. Körner F.: A new bound for the quadratic knapsack problem and its use in a branch and bound algorithm. Optimization. 17, 643–648 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  20. Körner F.: An efficient branch and bound algorithm to solve the quadratic integer programming problem. Computing. 30, 253–260 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  21. Liberti L., Maculan N.: Global Optimization: From Theory to Implementation, Chapter: Nonconvex Optimization and Its Applications. Springer, New York (2006)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Saxena A., Bonami P., Lee J.: Disjunctive Cuts for Non-Convex Mixed Integer Quadratically Constrained Programs. IPCO, Bologna (2008)

  24. Sherali H.D., Adams W.P.: A tight linearization and an algorithm for zero-one quadratic programming problems. Manage. Sci. 32(10), 1274–1290 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Tawarmalani M., Sahinidis N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99(3), 563–591 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Tawarmalani M., Sahinidis N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming. Kluwer, Dordrecht (2002)

    MATH  Google Scholar 

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Authors and Affiliations

  1. CEDRIC-ENSIIE, 1 square de la résistance, 91025, Evry cedex, France

    Alain Billionnet

  2. CEDRIC-CNAM, 292 rue Saint-Martin, 75141, Paris cedex 03, France

    Sourour Elloumi & Amélie Lambert

Authors
  1. Alain Billionnet
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  2. Sourour Elloumi
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  3. Amélie Lambert
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Correspondence to Amélie Lambert.

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Billionnet, A., Elloumi, S. & Lambert, A. Extending the QCR method to general mixed-integer programs. Math. Program. 131, 381–401 (2012). https://doi.org/10.1007/s10107-010-0381-7

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  • DOI: https://doi.org/10.1007/s10107-010-0381-7

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