Extending a CIP Framework to Solve MIQCPs

  • Timo BertholdEmail author
  • Stefan Heinz
  • Stefan Vigerske
Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 154)


This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP). The advantage of this approach is that we can utilize the full power of advanced MILP and CP technologies, in particular for the linear relaxation and the discrete components of the problem. We use an outer approximation generated by linearization of convex constraints and linear underestimation of nonconvex constraints to relax the problem. Further, we give an overview of the reformulation, separation, and propagation techniques that are used to handle the quadratic constraints efficiently. We implemented these methods in the branch-cut-and-price framework SCIP. Computational experiments indicating the potential of the approach and evaluating the impact of the algorithmic components are provided.

Key words

Mixed integer quadratically constrained programming constraint integer programming branch-and-cut convex relaxation domain propagation primal heuristic nonconvex 


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  1. 1.
    K. Abhishek, S. Leyffer, and J. Linderoth, FilMINT: An outer-approximation-based solver for nonlinear mixed integer programs, INFORMS Journal on Computing, 22 (2010), pp. 555–567.CrossRefMathSciNetGoogle Scholar
  2. 2.
    T. Achterberg, Constraint Integer Programming, PhD thesis, Technische Universit ¨at Berlin, 2007.Google Scholar
  3. 3.
    , SCIP: Solving Constraint Integer Programs, Math. Program. Comput.,1 (2009), pp. 1–41.Google Scholar
  4. 4.
    T. Achterberg, T. Berthold, T. Koch, and K. Wolter, Constraint integer programming: A new approach to integrate CP and MIP, in Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 5th International Conference, CPAIOR 2008, L. Perron and M. Trick, eds., Vol. 5015 of LNCS, Springer, 2008, pp. 6–20.Google Scholar
  5. 5.
    X. Bao, N. Sahinidis, and M. Tawarmalani, Multiterm polyhedral relaxations for nonconvex, quadratically-constrained quadratic programs, Optimization Methods and Software, 24 (2009), pp. 485–504.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    P. Belotti, J. Lee, L. Liberti, F. Margot, and A. W¨achter, Branching and bounds tightening techniques for non-convex MINLP, Optimization Methods and Software, 24 (2009), pp. 597–634.Google Scholar
  7. 7.
    A. Ben-Tal and A. Nemirovski, On polyhedral approximations of the second- order cone, Math. Oper. Res., 26 (2001), pp. 193–205.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    M. B´enichou, J. M. Gauthier, P. Girodet, G. Hentges, G. Ribi`ere, and O. Vincent, Experiments in mixed-integer linear programming, Math. Program., 1 (1971), pp. 76–94.Google Scholar
  9. 9.
    T. Berthold, Primal heuristics for mixed integer programs. Diploma thesis, Technische Universit¨at Berlin, 2006.Google Scholar
  10. 10.
    , RENS – relaxation enforced neighborhood search, ZIB-Report 07–28, Zuse Institute Berlin, 2007.Google Scholar
  11. 11.
    T. Berthold, S. Heinz, and M.E. Pfetsch, Nonlinear pseudo-boolean optimization: relaxation or propagation?, in Theory and Applications of Satisfiability Testing – SAT 2009, O. Kullmann, ed., no. 5584 in LNCS, Springer, 2009, pp. 441–446.Google Scholar
  12. 12.
    R.E. Bixby, M. Fenelon, Z. Gu, E. Rothberg, and R. Wunderling, MIP: theory and practice – closing the gap, in System Modelling and Optimization: Methods, Theory and Applications, M. Powell and S. Scholtes, eds., Kluwer, 2000, pp. 19–50.Google Scholar
  13. 13.
    P. Bonami, L.T. Biegler, A.R. Conn, G. Cornu´ejols, I.E. Grossmann, C.D. Laird, J. Lee, A. Lodi, F. Margot, N.W. Sawaya, and A. W¨achter, An algorithmic framework for convex mixed integer nonlinear programs, Discrete Optim., 5 (2008), pp. 186–204.Google Scholar
  14. 14.
    M.R. Bussieck, A.S. Drud, and A. Meeraus, MINLPLib - a collection of test models for mixed-integer nonlinear programming, INFORMS J. Comput., 15 (2003), pp. 114–119.CrossRefMathSciNetGoogle Scholar
  15. 15.
  16. 16.
    F. Domes and A. Neumaier, Quadratic constraint propagation, Constraints, 15 (2010), pp. 404–429.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    J.N. Hooker, Integrated Methods for Optimization, International Series in Operations Research & Management Science, Springer, New York, 2007.zbMATHGoogle Scholar
  18. 18.
    R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer, 1990.Google Scholar
  19. 19.
  20. 20.
    Y. Lin and L. Schrage, The global solver in the LINDO API, Optimization Methods and Software, 24 (2009), pp. 657–668.CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    G. McCormick, Computability of global solutions to factorable nonconvex programs: Part I-Convex Underestimating Problems, Math. Program., 10 (1976), pp. 147–175.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    H. Mittelmann, MIQP test instances.
  23. 23.
    MOSEK Corporation, The MOSEK optimization tools manual, 6.0 ed., 2009.Google Scholar
  24. 24.
    A. Saxena, P. Bonami, and J. Lee, Convex relaxations of non-convex mixed integer quadratically constrained programs: Projected formulations, Tech. Rep. RC24695, IBM Research, 2008. to appear in Math. Program.Google Scholar
  25. 25.
    M. Tawarmalani, J.-P.P. Richard, and K. Chung, Strong valid inequalities for orthogonal disjunctions and bilinear covering sets, Math. Program., 124 (2010), pp. 481–512.CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    M. Tawarmalani and N. Sahinidis, Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications, Kluwer Academic Publishers, 2002.Google Scholar
  27. 27.
    J.P. Vielma, S. Ahmed, and G.L. Nemhauser, A lifted linear programming branch-and-bound algorithm for mixed integer conic quadratic programs, INFORMS J. Comput., 20 (2008), pp. 438–450.CrossRefMathSciNetGoogle Scholar
  28. 28.
    A. W¨achter and L.T. Biegler, On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming, Math. Program., 106 (2006), pp. 25–57.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Zuse Institute BerlinBerlinGermany
  2. 2.Humboldt University BerlinBerlinGermany

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