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Algorithms and Software for Convex Mixed Integer Nonlinear Programs

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Mixed Integer Nonlinear Programming

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

Abstract

This paper provides a survey of recent progress and software for solving convex Mixed Integer Nonlinear Programs (MINLP)s, where the objective and constraints are defined by convex functions and integrality restrictions are imposed on a subset of the decision variables. Convex MINLPs have received sustained attention in recent years. By exploiting analogies to well-known techniques for solving Mixed Integer Linear Programs and incorporating these techniques into software, significant improvements have been made in the ability to solve these problems.

Supported by ANR grand BLAN06-1-138894.

The work of the second and third authors is supported by the US Department of Energy under grants DE-FG02-08ER25861 and DE-FG02-09ER25869, and the National Science Foundation under grant CCF-0830153.

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References

  1. K. Abhishek, S. Leyffer, and J.T. Linderoth, Feasibility pump heuristics for Mixed Integer Nonlinear Programs. Unpublished working paper, 2008.

    Google Scholar 

  2. , FilMINT: An outer-approximation-based solver for convex Mixed-Integer Nonlinear Programs, INFORMS Journal on Computing, 22, No. 4 (2010), pp. 555{567.

    Google Scholar 

  3. T. Achterberg and T. Berthold, Improving the feasibility pump, Technical Report ZIB-Report 05–42, Zuse Institute Berlin, September 2005.

    Google Scholar 

  4. T. Achterberg, T. Koch, and A. Martin, Branching rules revisited, Operations Research Letters, 33 (2004), pp. 42{54.

    Google Scholar 

  5. I. Akrotirianakis, I. Maros, and B. Rustem, An outer approximation based branch-and-cut algorithm for convex 0–1 MINLP problems, Optimization Methods and Software, 6 (2001), pp. 21{47.

    Google Scholar 

  6. D. Applegate, R. Bixby, V. Chv_atal, and W. Cook, On the solution of traveling salesman problems, in Documenta Mathematica Journal der Deutschen Mathematiker-Vereinigung, International Congress of Mathematicians, 1998, pp. 645{656.

    Google Scholar 

  7. A. Atamturk and V. Narayanan, Conic mixed integer rounding cuts, Mathematical Programming, 122 (2010), pp. 1{20.

    Google Scholar 

  8. E. Balas, Disjunctive programming, in Annals of Discrete Mathematics 5: Discrete Optimization, North Holland, 1979, pp. 3{51.

    Google Scholar 

  9. E. Balas, S. Ceria, and G. Corneujols, A lift-and-project cutting plane algorithm for ixed 0–1 programs, Mathematical Programming, 58 (1993), pp. 295{324.

    Google Scholar 

  10. E. Balas, S. Ceria, G. Cornu_ejols, and N. R. Natraj, Gomory cuts revisited, Operations Research Letters, 19 (1999), pp. 1{9.

    Google Scholar 

  11. E. Balas and M. Perregaard, Lift-and-project for mixed 0–1 programming: recent progress, Discrete Applied Mathematics, 123 (2002), pp. 129{154.

    Google Scholar 

  12. M.S. Bazaraa, H.D. Sherali, and C.M. Shetty, Nonlinear Programming: Theory and Algorithms, John Wiley and Sons, New York, second ed., 1993.

    Google Scholar 

  13. E.M.L. Beale, Branch and bound methods for mathematical programming systems, in Discrete Optimization II, P.L. Hammer, E.L. Johnson, and B.H.Korte, eds., North Holland Publishing Co., 1979, pp. 201{219.

    Google Scholar 

  14. E.W.L. Beale and J.A. Tomlin, Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables, in Proceedings of the 5th International Conference on Operations Research, J. Lawrence, ed., 1969, pp. 447{454.

    Google Scholar 

  15. A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization, SIAM, 2001. MPS/SIAM Series on Optimization.

    Google Scholar 

  16. J.F. Benders, Partitioning procedures for solving mixed variable programming problems, Numerische Mathematik, 4 (1962), pp. 238{252.

    Google Scholar 

  17. M. B_enichou, J.M. Gauthier, P. Girodet, G. Hentges, G. Ribi_ere, and O. Vincent, Experiments in Mixed-Integer Linear Programming, Mathematical Programming, 1 1971), pp. 76{94.

    Google Scholar 

  18. L. Bertacco, M. Fischetti, and A. Lodi, A feasibility pump heuristic for general mixed-integer problems, Discrete Optimization, 4 (2007), pp. 63{76.

    Google Scholar 

  19. T. Berthold, Primal Heuristics for Mixed Integer Programs, Master's thesis, Technische Universitat Berlin, 2006.

    Google Scholar 

  20. T. Berthold and A. Gleixner, Undercover - a primal heuristic for MINLP based on sub-mips generated by set covering, Tech. Rep. ZIB-Report 09–40, Konrad-Zuse-Zentrum fur Informationstechnik Berlin (ZIB), 2009.

    Google Scholar 

  21. D. Bienstock, Computational study of a family of mixed-integer quadratic programming problems, Mathematical Programming, 74 (1996), pp. 121{140.

    Google Scholar 

  22. R. Bixby and E. Rothberg, Progress in computational mixed integer programming. A look back from the other side of the tipping point, Annals of Operations Research, 149 (2007), pp. 37{41.

    Google Scholar 

  23. P. Bonami, Branching strategies and heuristics in branch-and-bound for convex MINLPs, November 2008. Presentation at IMA Hot Topics Workshop: Mixed-Integer Nonlinear Optimization: Algorithmic Advances and Applications.

    Google Scholar 

  24. P. Bonami, L.T. Biegler, A.R. Conn, G. Cornu_ejols, I.E. Grossmann, C.D. Laird, J. Lee, A. Lodi, F. Margot, N. Sawaya, and A. Wachter, An algorithmic framework for convex Mixed Integer Nonlinear Programs, Discrete Optimization, 5 (2008), pp. 186{204.

    Google Scholar 

  25. P. Bonami, G. Cornu_ejols, A. Lodi, and F. Margot, A feasibility pump for Mixed Integer Nonlinear Programs, Mathematical Programming, 119 (2009), pp. 331{352.

    Google Scholar 

  26. P. Bonami and J. Gonc_alves, Heuristics for convex mixed integer nonlinear programs, Computational Optimization and Applications. To appear DOI: 10.1007/s10589-010-9350-6.

  27. R. Boorstyn and H. Frank, Large-scale network topological optimization, IEEE Transactions on Communications, 25 (1977), pp. 29{47.

    Google Scholar 

  28. B. Borchers and J.E. Mitchell, An improved branch and bound algorithm for Mixed Integer Nonlinear Programs, Computers & Operations Research, 21 (1994), pp. 359{368.

    Google Scholar 

  29. , A computational comparison of branch and bound and outer approximation algorithms for 0–1 Mixed Integer Nonlinear Programs, Computers & Operations Research, 24 (1997), pp. 699{701.

    Google Scholar 

  30. M.R. Bussieck and A. Drud, Sbb: A new solver for Mixed Integer Nonlinear Programming, talk, OR 2001, Section "Continuous Optimization", 2001.

    Google Scholar 

  31. M.R. Bussieck, A.S. Drud, and A. Meeraus, MINLPLib { a collection of test models for Mixed-Integer Nonlinear Programming, INFORMS Journal on Computing, 15 (2003).

    Google Scholar 

  32. R.H. Byrd, J. Nocedal, and R.A. Waltz, KNITRO: An integrated package for nonlinear optimization, in Large Scale Nonlinear Optimization, Springer Verlag, 2006, pp. 35{59.

    Google Scholar 

  33. I. Castillo, J. Westerlund, S. Emet, and T. Westerlund, Optimization of block layout deisgn problems with unequal areas: A comparison of milp and minlp optimization methods, Computers and Chemical Engineering, 30 (2005), pp. 54{69.

    Google Scholar 

  34. M.T. Cezik and G. Iyengar, Cuts for mixed 0–1 conic programming, Mathematical Programming, 104 (2005), pp. 179{202.

    Google Scholar 

  35. V. Chv_atal, Edmonds polytopes and a heirarchy of combinatorial problems, Discrete Mathematics, 4 (1973), pp. 305{337.

    Google Scholar 

  36. M. Conforti, G. Cornu_ejols, and G. Zambelli, Polyhedral approaches to Mixed Integer Linear Programming, in 50 Years of Integer Programming 1958–2008, M. Junger, T. Liebling, D. Naddef, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey, eds., Springer, 2009.

    Google Scholar 

  37. G. Cornu_ejols, L. Liberti, and G. Nannicini, Improved strategies for branching on general disjunctions. To appear in Mathematical Programming A. DOI: 10.1007/s10107-009-0333-2.

  38. R.J. Dakin, A tree search algorithm for mixed programming problems, Computer Journal, 8 (1965), pp. 250{255.

    Google Scholar 

  39. E. Danna, E. Rothberg, and C. LePape, Exploring relaxation induced neighborhoods to improve MIP solutions, Mathematical Programming, 102 (2005), pp. 71{90.

    Google Scholar 

  40. E. Dolan and J. Mor_e, Benchmarking optimization software with performance pro_les, Mathematical Programming, 91 (2002), pp. 201{213.

    Google Scholar 

  41. S. Drewes, Mixed Integer Second Order Cone Programming, PhD thesis, Technische Universitat Darmstadt, 2009.

    Google Scholar 

  42. A. S. Drud, CONOPT { a large-scale GRG code, ORSA Journal on Computing, 6 (1994), pp. 207{216.

    Google Scholar 

  43. M.A. Duran and I. Grossmann, An outer-approximation algorithm for a class of Mixed-Integer Nonlinear Programs, Mathematical Programming, 36 (1986), pp. 307{339.

    Google Scholar 

  44. J. Eckstein, Parallel branch-and-bound algorithms for general mixed integer programming on the CM-5, SIAM Journal on Optimization, 4 (1994), pp. 794{

    Google Scholar 

  45. 814.

    Google Scholar 

  46. S. Elhedhli, Service System Design with Immobile Servers, Stochastic Demand, and Congestion, Manufacturing & Service Operations Management, 8 (2006), pp. 92{97.

    Google Scholar 

  47. A. M. Eliceche, S. M. Corval_an, and P. Mart__nez, Environmental life cycle impact as a tool for process optimisation of a utility plant, Computers and Chemical Engineering, 31 (2007), pp. 648{656.

    Google Scholar 

  48. Fair Isaac Corporation, XPRESS-MP Reference Manual, 2009. Release 2009.

    Google Scholar 

  49. M. Fischetti, F. Glover, and A. Lodi, The feasibility pump, Mathematical Programming, 104 (2005), pp. 91{104.

    Google Scholar 

  50. M. Fischetti and A. Lodi, Local branching, Mathematical Programming, 98 (2003), pp. 23{47.

    Google Scholar 

  51. M. Fischetti and D. Salvagnin, Feasibility pump 2.0, Tech. Rep., University of Padova, 2008.

    Google Scholar 

  52. R. Fletcher and S. Leyffer, Solving Mixed Integer Nonlinear Programs by outer approximation, Mathematical Programming, 66 (1994), pp. 327{349.

    Google Scholar 

  53. , User manual for _lterSQP, 1998. University of Dundee Numerical Analysis Report NA-181.

    Google Scholar 

  54. A. Flores-Tlacuahuac and L.T. Biegler, Simultaneous mixed-integer dynamic optimization for integrated design and control, Computers and Chemical Engineering, 31 (2007), pp. 648{656.

    Google Scholar 

  55. J.J.H. Forrest, J.P.H. Hirst, and J.A. Tomlin, Practical solution of large scale mixed integer programming problems with UMPIRE, Management Science, 20 (1974), pp. 736{773.

    Google Scholar 

  56. M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W.H. Freeman and Company, New York, 1979.

    Google Scholar 

  57. A. Geoffrion, Generalized Benders decomposition, Journal of Optimization Theory and Applications, 10 (1972), pp. 237{260.

    Google Scholar 

  58. P.E. Gill, W. Murray, and M.A. Saunders, SNOPT: An SQP algorithm for large{scale constrained optimization, SIAM Journal on Optimization, 12 (2002), pp. 979{1006.

    Google Scholar 

  59. R.E. Gomory, Outline of an algorithm for integer solutions to linear programs, Bulletin of the American Mathematical Monthly, 64 (1958), pp. 275{278.

    Google Scholar 

  60. , An algorithm for the mixed integer problem, Tech. Rep. RM-2597, The RAND Corporation, 1960.

    Google Scholar 

  61. I. Grossmann, J. Viswanathan, A.V.R. Raman, and E. Kalvelagen, GAMS/DICOPT: A discrete continuous optimization package, Math. Methods Appl. Sci, 11 (2001), pp. 649{664.

    Google Scholar 

  62. O. Gunluk, J. Lee, and R. Weismantel, MINLP strengthening for separaable convex quadratic transportation-cost u, Tech. Rep. RC24213 (W0703-042), IBM Research Division, March 2007.

    Google Scholar 

  63. O.K. Gupta and A. Ravindran, Branch and bound experiments in convex non- linear integer programming, Management Science, 31 (1985), pp. 1533{1546.

    Google Scholar 

  64. Gurobi Optimization, Gurobi Optimizer Reference Manual, 2009. Version 2.

    Google Scholar 

  65. I. Harjunkoski, R.Porn, and T. Westerlund, MINLP: Trim-loss problem, in Encyclopedia of Optimization, C.A. Floudas and P.M. Pardalos, eds., Springer, 2009, pp. 2190{2198.

    Google Scholar 

  66. J.-B. Hiriart-Urruty and C. Lemarechal, Convex Analysis and Minimization Algorithms I: Fundamentals (Grundlehren Der Mathematischen Wis-senschaften), Springer, October 1993.

    Google Scholar 

  67. IBM, Using the CPLEX Callable Library, Version 12, 2009.

    Google Scholar 

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

    Google Scholar 

  69. N.J. Jobst, M.D. Horniman, C.A. Lucas, and G. Mitra, Computational as- pects of alternative portfolio selection models in the presence of discrete asset choice constraints, Quantitative Finance, 1 (2001), pp. 489{501.

    Google Scholar 

  70. M. Karamanov and G. Cornu_ejols, Branching on general disjunctions, tech. rep., Carnegie Mellon University, July 2005. Revised August 2009. Available at http://integer.tepper.cmu.edu.

  71. J.E. Kelley, The cutting plane method for solving convex programs, Journal of SIAM, 8 (1960), pp. 703{712.

    Google Scholar 

  72. M. K_l_nc_, J. Linderoth, J. Luedtke, and A. Miller, Disjunctive strong branching inequalities for Mixed Integer Nonlinear Programs.

    Google Scholar 

  73. G.R. Kocis and I.E. Grossmann, Relaxation strategy for the structural optimization of process owheets, Industrial Engineering Chemical Research, 26 (1987), pp. 1869{1880.

    Google Scholar 

  74. C.D. Laird, L.T. Biegler, and B. van Bloemen Waanders, A mixed integer approach for obtaining unique solutions in source inversion of drinking water networks, Journal of Water Resources Planning and Management, Special Issue on Drinking Water Distribution Systems Security, 132 (2006), pp. 242{251.

    Google Scholar 

  75. A.H. Land and A.G. Doig, An automatic method for solving discrete programming problems, Econometrica, 28 (1960), pp. 497{520.

    Google Scholar 

  76. M. Lejeune, A uni_ed approach for cycle service levels, Tech. Rep. George Washington University, 2009. Available on Optimization Online http://www.optimization-online.org/DB HTML/2008/11/2144.html.

  77. S. Leyffer, Deterministic Methods for Mixed Integer Nonlinear Programming, PhD thesis, University of Dundee, Dundee, Scotland, UK, 1993.

    Google Scholar 

  78. , User manual for MINLP-BB, 1998. University of Dundee.

    Google Scholar 

  79. , Integrating SQP and branch-and-bound for Mixed Integer Nonlinear Programming, Computational Optimization & Applications, 18 (2001), pp. 295{309.

    Google Scholar 

  80. , MacMINLP: Test problems for Mixed Integer Nonlinear Programming, 2003. http://www.mcs.anl.gov/~leyffer/macminlp.

  81. , Nonlinear branch-and-bound revisited, August 2009. Presentation at 20th International Symposium on Mathematical Programming.

    Google Scholar 

  82. L. Liberti, Reformulations in mathematical programming: Symmetry, Mathematical Programming (2010). To appear.

    Google Scholar 

  83. J. Linderoth and T. Ralphs, Noncommercial software for Mixed-Integer Linear Programming, in Integer Programming: Theory and Practice, CRC Press Operations Research Series, 2005, pp. 253{303.

    Google Scholar 

  84. J.T. Linderoth and M.W.P. Savelsbergh, A computational study of search strategies in mixed integer programming, INFORMS Journal on Computing, 11 (1999), pp. 173{187.

    Google Scholar 

  85. A. Lodi, MIP computation and beyond, in 50 Years of Integer Programming 1958–2008, M. Junger, T. Liebling, D. Naddef, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey, eds., Springer, 2009.

    Google Scholar 

  86. H. Marchand and L.A. Wolsey, Aggregation and mixed integer rounding to solve MIPs, Operations Research, 49 (2001), pp. 363{371.

    Google Scholar 

  87. F. Margot, Exploiting orbits in symmetric ILP, Mathematical Programming, Series B, 98 (2003), pp. 3{21.

    Google Scholar 

  88. R.D. McBride and J.S. Yormark, An implicit enumeration algorithm for quadratic integer programming, Management Science, 26 (1980), pp. 282{296.

    Google Scholar 

  89. Mosek ApS, 2009. www.mosek.com.

  90. B. Murtagh and M. Saunders, MINOS 5.4 user's guide, Report SOL 83-20R, Department of Operations Research, Stanford University, 1993.

    Google Scholar 

  91. K.G. Murty and S.N. Kabadi, Some NP-complete problems in quadratic and nonlinear programming, Mathematical Programming, 39 (1987), pp. 117{ 129.

    Google Scholar 

  92. S. Nabal and L. Schrage, Modeling and solving nonlinear integer programming problems. Presented at Annual AIChE Meeting, Chicago, 1990.

    Google Scholar 

  93. G. Nemhauser and L.A. Wolsey, Integer and Combinatorial Optimization, John Wiley and Sons, New York, 1988.

    Google Scholar 

  94. G.L. Nemhauser, M.W.P. Savelsbergh, and G.C. Sigismondi, MINTO, a Mixed INTeger Optimizer, Operations Research Letters, 15 (1994), pp. 47{58.

    Google Scholar 

  95. J. Nocedal and S.J. Wright, Numerical Optimization, Springer-Verlag, New York, second ed., 2006.

    Google Scholar 

  96. J. Ostrowski, J. Linderoth, F. Rossi, and S. Smriglio, Orbital branching, Mathematical Programming, 126 (2011), pp. 147{178.

    Google Scholar 

  97. I. Quesada and I.E. Grossmann, An LP/NLP based branch{and{bound algorithm for convex MINLP optimization problems, Computers and Chemical Engineering, 16 (1992), pp. 937{947.

    Google Scholar 

  98. D.E. Ravemark and D.W.T. Rippin, Optimal design of a multi-product batch plant, Computers & Chemical Engineering, 22 (1998), pp. 177 { 183.

    Google Scholar 

  99. N. Sawaya, Reformulations, relaxations and cutting planes for generalized disjunctive programming, PhD thesis, Chemical Engineering Department, Carnegie Mellon University, 2006.

    Google Scholar 

  100. N. Sawaya, C.D. Laird, L.T. Biegler, P. Bonami, A.R. Conn, G. Cornu_ejols, I. E. Grossmann, J. Lee, A. Lodi, F. Margot, and A. Wachter, CMU-IBM open source MINLP project test set, 2006. http://egon.cheme.cmu.edu/ibm/page.htm.

  101. A. Schrijver, Theory of Linear and Integer Programming, Wiley, Chichester, 1986.

    Google Scholar 

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

    Google Scholar 

  103. M. Turkay and I.E. Grossmann, Logic-based minlp algorithms for the optimal synthesis of process networks, Computers & Chemical Engineering, 20 (1996), pp. 959 { 978.

    Google Scholar 

  104. R.J. Vanderbei, LOQO: An interior point code for quadratic programming, Optimization Methods and Software (1998).

    Google Scholar 

  105. A. Vecchietti and I.E. Grossmann, LOGMIP: a disjunctive 0–1 non-linear optimizer for process system models, Computers and Chemical Engineering, 23 (1999), pp. 555 { 565.

    Google Scholar 

  106. J. Viswanathan and I.E. Grossmann, A combined penalty function and outer{approximation method for MINLP optimization, Computers and Chemical Engineering, 14 (1990), pp. 769{782.

    Google Scholar 

  107. A. Wachter, Some recent advanced in Mixed-Integer Nonlinear Programming, May 2008. Presentation at the SIAM Conference on Optimization.

    Google Scholar 

  108. A. Wachter and L.T. Biegler, On the implementation of a primal-dual interior point _lter line search algorithm for large-scale nonlinear programming, Mathematical Programming, 106 (2006), pp. 25{57.

    Google Scholar 

  109. R. Waltz, Current challenges in nonlinear optimization, 2007. Presentation at San Diego Supercomputer Center: CIEG Spring Orientation Workshop, available at www.sdsc.edu/us/training/workshops/2007sac_ studentworkshop/docs/SDSC07.ppt.

  110. T. Westerlund, H.I. Harjunkoski, and R. Porn, An extended cutting plane method for solving a class of non-convex minlp problems, Computers and Chemical Engineering, 22 (1998), pp. 357{365.

    Google Scholar 

  111. T. Westerlund and K. Lundqvist, Alpha-ECP, version 5.101. an interactive minlp-solver based on the extended cutting plane method, in Updated version of Report 01-178-A, Process Design Laboratory, Abo Akademi Univeristy, 2005.

    Google Scholar 

  112. T. Westerlund and F. Pettersson, A cutting plane method for solving convex MINLP problems, Computers and Chemical Engineering, 19 (1995), pp. s131{s136.

    Google Scholar 

  113. T. Westerlund and R. Porn,. a cutting plane method for minimizing pseudconvex functions in the mixed integer case, Computers and Chemical Engineering, 24 (2000), pp. 2655{2665.

    Google Scholar 

  114. L.A. Wolsey, Integer Programming, John Wiley and Sons, New York, 1998.

    Google Scholar 

  115. Y. Zhu and T. Kuno, A disjunctive cutting-plane-based branch-and-cut algorithm for 0–1 mixed-integer convex nonlinear programs, Industrial and Engineering Chemistry Research, 45 (2006), pp. 187{196.

    Google Scholar 

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Bonami, P., Kilinç, M., Linderoth, J. (2012). Algorithms and Software for Convex Mixed Integer Nonlinear Programs. 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_1

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