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Mixed Integer Nonlinear Programming pp 117–144Cite as

Disjunctive Cuts for Nonconvex MINLP

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 154))

Abstract

Mixed Integer Nonlinear Programming (MINLP) problems present two main challenges: the integrality of a subset of variables and nonconvex (nonlinear) objective function and constraints. Many exact solvers for MINLP are branch-and-bound algorithms that compute a lower bound on the optimal solution using a linear programming relaxation of the original problem. In order to solve these problems to optimality, disjunctions can be used to partition the solution set or to obtain strong lower bounds on the optimal solution of the problem. In the MINLP context, the use of disjunctions for branching has been subject to intense research, while the practical utility of disjunctions as a means of generating valid linear inequalities has attracted some attention only recently. We describe an application to MINLP of a well-known separation method for disjunctive cuts that has shown to be very effective in Mixed Integer Linear Programming(MILP). As the experimental results show, this application obtains encouraging results in the MINLP case even when a simple separation method is used.

AMS(MOS) subject classifications. 90C57.

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References

  1. T. Achterberg, T. Koch, and A. Martin, Branching rules revisited, OR Letters, 33 (2005), pp. 42–54.

    MATH  MathSciNet  Google Scholar 

  2. K. Andersen, G. Cornu´ejols, and Y. Li, Split closure and intersection cuts, Mathematical Programming, 102 (2005), pp. 457–493.

    Google Scholar 

  3. K.M. Anstreicher, Semidefinite programming versus the reformulation- linearization technique for nonconvex quadratically constrained quadratic programming, Journal of Global Optimization, 43 (2009), pp. 471–484.

    Article  MATH  MathSciNet  Google Scholar 

  4. D.L. Applegate, R.E. Bixby, V. Chv´atal, andW.J. Cook, The Traveling Sales- man Problem, A Computational Study, Princeton, 2006.

    Google Scholar 

  5. E. Balas, Disjunctive programming: Properties of the convex hull of feasible points, Discrete Applied Mathematics, 89 (1998), pp. 3–44.

    Article  MATH  MathSciNet  Google Scholar 

  6. E. Balas and P. Bonami, New variants of Lift-and-Project cut generation from the LP tableau: Open Source implementation and testing, in Integer Programming and Combinatorial Optimization, Vol. 4513 of Lecture Notes in Computer Science, Springer Berlin/Heidelberg, 2007, pp. 89–103.

    Google Scholar 

  7. E. Balas, S. Ceria, and G. Cornu´ejols, A lift-and-project cutting plane algorithm for mixed 0–1 programs, Mathematical Programming, 58 (1993), pp. 295–324.

    Google Scholar 

  8. , Mixed 0–1 programming by lift-and-project in a branch-and-cut framework, Management Science, 42 (1996), pp. 1229–1246.

    Google Scholar 

  9. E. Balas and M. Perregaard, A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer Gomory cuts for 0–1 programming, Mathematical Programming, 94 (2003), pp. 221–245.

    Article  MATH  MathSciNet  Google Scholar 

  10. E. Balas and A. Saxena, Optimizing over the split closure, Mathematical Programming, Series A, 113 (2008), pp. 219–240.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Belotti, couenne: a user’s manual, tech. rep., Lehigh University, 2009.

    Google Scholar 

  12. 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 

  13. M. Benichou, J.M. Gauthier, P. Girodet, G. Hentges, G. Ribiere, and O. Vincent, Experiments in mixed–integer linear programming, Mathematical Programming, 1 (1971), pp. 76–94.

    Article  MATH  MathSciNet  Google Scholar 

  14. L.T. Biegler, I.E. Grossmann, and A.W. Westerberg, Systematic Methods of Chemical Process Design, Prentice Hall, Upper Saddle River (NJ), 1997.

    Google Scholar 

  15. S. Burer and D. Vandenbussche, Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound, Comput. Optim. Appl., 43 (2009), pp. 181–195.

    Article  MATH  MathSciNet  Google Scholar 

  16. M.R. Bussieck, A.S. Drud, and A. Meeraus, MINLPLib – a collection of test models for mixed-integer nonlinear programming, INFORMS Journal of Computing, 15 (2003), pp. 114–119. Available online at http://www.gamsworld.org/minlp/minlplib/minlpstat.htm.

  17. A. Caprara and A.N. Letchford, On the separation of split cuts and related inequalities, Mathematical Programming, Series B, 94 (2003), pp. 279–294.

    Article  MATH  MathSciNet  Google Scholar 

  18. CBC-2.1, Available from https://projects.coin-or.org/cbc.

  19. M.T. C¸ ezik and G. Iyengar, Cuts for Mixed 0–1 Conic Programming, Mathematical Programming, Ser. A, 104 (2005), pp. 179–200.

    Google Scholar 

  20. CglLandP, https://projects.coin-or.org/Cgl/wiki/CglLandP.

  21. J.S. Cohen, Computer algebra and symbolic computation: elementary algorithms, A.K. Peters, Ltd., 2002.

    Google Scholar 

  22. W. Cook, R. Kannan, and A. Schrijver, Chv´atal closures for mixed integer programming problems, Mathematical Programming, 47 (1990), pp. 155–174.

    Article  MATH  MathSciNet  Google Scholar 

  23. Couenne, https://www.coin-or.org/Couenne.

  24. E.D. Dolan and J.J. Mor´e, Benchmarking optimization software with performance profiles, Mathematical Programming, 91 (2002), pp. 201–213.

    Google Scholar 

  25. S. Drewes, Mixed Integer Second Order Cone Programming, PhD thesis, Technische Universit¨at Darmstadt, 2009.

    Google Scholar 

  26. C.A. Floudas, Global optimization in design and control of chemical process systems, Journal of Process Control, 10 (2001), pp. 125–134.

    Article  Google Scholar 

  27. A. Frangioni and C. Gentile, Perspective cuts for a class of convex 0–1 mixed integer programs, Mathematical Programming, 106 (2006), pp. 225–236.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. F¨ugenschuh and L. Schewe, Solving a nonlinear mixed-integer sheet metal design problem with linear approximations, work in progress.

    Google Scholar 

  29. GAMS Development Corp., Gamsworld global optimization library. http://www.gamsworld.org/global/globallib/globalstat.htm.

  30. E. Hansen, Global Optimization Using Interval Analysis, Marcel Dekker, Inc., New York, 1992.

    MATH  Google Scholar 

  31. C. Helmberg and F. Rendl, Solving quadratic (0,1)-problems by semidefinite programs and cutting planes, Mathematical Programming, 82 (1998), pp. 291–315.

    MATH  MathSciNet  Google Scholar 

  32. R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer Verlag, Berlin, 1996.

    MATH  Google Scholar 

  33. R.C. Jeroslow, There cannot be any algorithm for integer programming with quadratic constraints, Operations Research, 21 (1973), pp. 221–224.

    Article  MATH  MathSciNet  Google Scholar 

  34. J.J. J´udice, H.D. Sherali, I.M. Ribeiro, and A.M. Faustino, A complementarity-based partitioning and disjunctive cut algorithm for mathe-matical programming problems with equilibrium constraints, Journal of Global Optimization, 36 (2006), pp. 89–114.

    Google Scholar 

  35. J. Kallrath, Solving planning and design problems in the process industry using mixed integer and global optimization, Annals of Operations Research, 140 (2005), pp. 339–373.

    Article  MATH  MathSciNet  Google Scholar 

  36. M. Karamanov and G. Cornu´ejols, Branching on general disjunctions, Mathematical Programming (2007).

    Google Scholar 

  37. S. Leyffer, MacMINLP: ampl collection of MINLPs. Available at http://www-unix.mcs.anl.gov/~leyffer/MacMINLP.

  38. L. Liberti, Writing global optimization software, in Global Optimization: from Theory to Implementation, L. Liberti and N. Maculan, eds., Springer, Berlin, 2006, pp. 211–262.

    Google Scholar 

  39. L. Liberti, C. Lavor, and N. Maculan, A branch-and-prune algorithm for the molecular distance geometry problem, International Transactions in Operational Research, 15 (2008), pp. 1–17.

    Article  MATH  MathSciNet  Google Scholar 

  40. L. Liberti, C. Lavor, M.A.C. Nascimento, and N. Maculan, Reformulation in mathematical programming: an application to quantum chemistry, Discrete Applied Mathematics, 157 (2009), pp. 1309–1318.

    Article  MATH  MathSciNet  Google Scholar 

  41. L. Liberti and C.C. Pantelides, Convex envelopes of monomials of odd degree, Journal of Global Optimization, 25 (2003), pp. 157–168.

    Article  MATH  MathSciNet  Google Scholar 

  42. Lindo Systems, Lindo solver suite. Available online at http://www.gams.com/solvers/lindoglobal.pdf.

  43. M.L. Liu, N.V. Sahinidis, and J.P. Shectman, Planning of chemical process networks via global concave minimization, in Global Optimization in Engineering Design, I. Grossmann, ed., Springer, Boston, 1996, pp. 195–230.

    Google Scholar 

  44. R. Lougee-Heimer, The Common Optimization INterface for Operations Research, IBM Journal of Research and Development, 47 (2004), pp. 57–66.

    Article  Google Scholar 

  45. R. Lougee-Heimer, Cut generation library. http://projects.coin-or.org/Cgl,2006.

  46. G.P. McCormick, Computability of global solutions to factorable nonconvex programs: Part I — Convex underestimating problems, Mathematical Programming,10 (1976), pp. 146–175.

    Google Scholar 

  47. H. Mittelmann, A collection of mixed integer quadratically constrained quadratic programs. http://plato.asu.edu/ftp/ampl files/miqp ampl.

  48. R.E. Moore, Methods and Applications of Interval Analysis, Siam, Philadelphia, 1979.

    Book  MATH  Google Scholar 

  49. J.H. Owen and S. Mehrotra, A disjunctive cutting plane procedure for general mixed-integer linear programs, Mathematical Programming, 89 (2001), pp. 437–448.

    Article  MATH  MathSciNet  Google Scholar 

  50. A.T. Phillips and J.B. Rosen, A quadratic assignment formulation of the molecular conformation problem, tech. rep., CSD, Univ. of Minnesota, 1998.

    Google Scholar 

  51. I. Quesada and I.E. Grossmann, Global optimization of bilinear process networks and multicomponent flows, Computers & Chemical Engineering, 19 (1995), pp. 1219–1242.

    Article  Google Scholar 

  52. H. Ratschek and J. Rokne, Interval methods, in Handbook of Global Optimization, R. Horst and P. M. Pardalos, eds., Vol. 1, Kluwer Academic Publishers, Dordrecht, 1995, pp. 751–828.

    Google Scholar 

  53. F. Rendl and R. Sotirov, Bounds for the quadratic assignment problem using the bundle method, Mathematical Programming, 109 (2007), pp. 505–524.

    Article  MATH  MathSciNet  Google Scholar 

  54. N.V. Sahinidis, BARON: A general purpose global optimization software package, Journal of Global Optimization, 8 (1996), pp. 201–205.

    Article  MATH  MathSciNet  Google Scholar 

  55. A. Saxena, P. Bonami, and J. Lee, Disjunctive cuts for non-convex mixed integer quadratically constrained programs, in Proceedings of the 13th Integer Programming and Combinatorial Optimization Conference, A. Lodi, A. Panconesi, and G. Rinaldi, eds., Vol. 5035 of Lecture Notes in Computer Science, 2008, pp. 17–33.

    Google Scholar 

  56. , Convex relaxations of non-convex mixed integer quadratically constrained programs: Projected formulations, Mathematical Programming (2011). To appear.

    Google Scholar 

  57. H. Scheel and S. Scholtes, Mathematical programs with complementarity constraints: Stationarity, optimality, and sensitivity, Mathematics of Operations Research, 25 (2000), pp. 1–22.

    Article  MATH  MathSciNet  Google Scholar 

  58. E.M.B. Smith, On the Optimal Design of Continuous Processes, PhD thesis, Imperial College of Science, Technology and Medicine, University of London, Oct. 1996.

    Google Scholar 

  59. E.M.B. Smith and C.C. Pantelides, A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs, Computers & Chem. Eng., 23 (1999), pp. 457–478.

    Article  Google Scholar 

  60. R.A. Stubbs and S. Mehrotra, A branch-and-cut method for 0–1 mixed convex programming, Mathematical Programming, 86 (1999), pp. 515–532.

    Article  MATH  MathSciNet  Google Scholar 

  61. M. Tawarmalani and N.V. Sahinidis, Convexification and global optimization in continuous and mixed-integer nonlinear programming: Theory, algorithms, software and applications, Vol. 65 of Nonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, 2002.

    Google Scholar 

  62. , Global optimization of mixed-integer nonlinear programs: A theoretical and computational study, Mathematical Programming, 99 (2004), pp. 563–591.

    Google Scholar 

  63. D. Vandenbussche and G.L. Nemhauser, A branch-and-cut algorithm for non-convex quadratic programs with box constraints, Mathematical Programming, 102 (2005), pp. 559–575.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA

    Pietro Belotti

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  1. Pietro Belotti
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  1. , College of Engineering, University of Michigan, 2355 Bonisteel Boulevard, Ann Arbor, 48109, Michigan, USA

    Jon Lee

  2. , Mathematics and Computer Science, Argonne National Laboratory, Argonne, 60439, Illinois, USA

    Sven Leyffer

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Belotti, P. (2012). Disjunctive Cuts for Nonconvex MINLP. In: Lee, J., Leyffer, S. (eds) Mixed Integer Nonlinear Programming. The IMA Volumes in Mathematics and its Applications, vol 154. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1927-3_5

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