Optimization and Engineering

, Volume 3, Issue 3, pp 227–252 | Cite as

Review of Nonlinear Mixed-Integer and Disjunctive Programming Techniques

  • Ignacio E. Grossmann


This paper has as a major objective to present a unified overview and derivation of mixed-integer nonlinear programming (MINLP) techniques, Branch and Bound, Outer-Approximation, Generalized Benders and Extended Cutting Plane methods, as applied to nonlinear discrete optimization problems that are expressed in algebraic form. The solution of MINLP problems with convex functions is presented first, followed by a brief discussion on extensions for the nonconvex case. The solution of logic based representations, known as generalized disjunctive programs, is also described. Theoretical properties are presented, and numerical comparisons on a small process network problem.

mixed-integer programming disjunctive programming nonlinear programming 


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  1. C. S. Adjiman, I. P. Androulakis, and C. A. Floudas, “Global optimization of mixed-integer nonlinear problems,” AIChE Journal vol. 16, no. 9, pp. 1769–1797, 2000.Google Scholar
  2. E. Balas, “Disjunctive programming and a hierarchy of relaxations for discrete optimization problems,” SIAM J. Alg. Disc. Meth. vol. 6, pp. 466–486, 1985.Google Scholar
  3. E. Balas, S. Ceria, and G. Cornuejols, “A lift-and-project cutting plane algorithm for mixed 0–1 programs,” Mathematical Programming vol. 58, pp. 295–324, 1993.Google Scholar
  4. C. Barnhart, E. L. Johnson, G. L. Nemhauser, M.W. P. Savelsbergh, and P. H. Vance, “Branch-and-price: Column generation for solving huge integer programs,” Operations Research vol. 46, pp. 316–329, 1998.Google Scholar
  5. M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming, John Wiley: New York, 1994.Google Scholar
  6. N. Beaumont, “An algorithm for disjunctive programs,” European Journal of Operations Research vol. 48, pp. 362–371, 1991.Google Scholar
  7. J. F. Benders, “Partitioning procedures for solving mixed-variables programming problems,” Numeri. Math. vol. 4, pp. 238–252, 1962.Google Scholar
  8. L. T. Biegler, I. E. Grossmann, and A.W. Westerberg, Systematic Methods for Chemical Process Design, Prentice-Hall: Englewood Cliffs, NJ, 1997.Google Scholar
  9. B. Borchers and J. E. Mitchell, “An improved branch and bound algorithm for mixed integer nonlinear programming,” Computers and Operations Research vol. 21, pp. 359–367, 1994.Google Scholar
  10. A. Brooke, D. Kendrick, A. Meeraus, and R. Raman, “GAMS–Auser's guide,” available at, 1998.Google Scholar
  11. S. Ceria and J. Soares, “Convex programming for disjunctive optimization,” Mathematical Programming vol. 86, no. 3, pp. 595–614, 1999.Google Scholar
  12. R. J. Dakin, “A tree search algorithm for mixed-integer programming problems,” Computer Journal vol. 8, pp. 250–255, 1965.Google Scholar
  13. Ding-Mei and R. W. H. Sargent, “A combined SQP and branch and bound algorithm for MINLP optimization,” Internal Report, Centre for Process Systems Engineering, London, 1992.Google Scholar
  14. M. A. Duran and I. E. Grossmann, “An outer-approximation algorithm for a class of mixed-integer nonlinear programs,” Math Programming vol. 36, p. 307, 1986.Google Scholar
  15. J. E. Falk and R. M. Soland, “An algorithm for separable nonconvex programming problems,” Management Science vol. 15, pp. 550–569, 1969.Google Scholar
  16. R. Fletcher and S. Leyffer, “Solving mixed integer nonlinear programs by outer approximation,” Math Programming vol. 66, p. 327, 1994.Google Scholar
  17. O. E. Flippo and A. H. G. Rinnoy Kan, “Decomposition in general mathematical programming,” Mathematical Programming vol. 60, pp. 361–382, 1993.Google Scholar
  18. C. A. Floudas, Deterministic Global Optimization: Theory, Methods and Applications, Kluwer Academic Publishers: Dordrecht, The Netherlands, 2000.Google Scholar
  19. A. M. Geoffrion, “Generalized benders decomposition,” Journal of Optimization Theory and Applications vol. 10, no. 4, pp. 237–260, 1972.Google Scholar
  20. I. E. Grossmann, “Mixed-integer optimization techniques for algorithmic process synthesis,” Advances in Chemical Engineering, J. L. Anderson, ed., vol. 23: Process Synthesis, pp. 171–246, Academic Press: San Diego, 1996a.Google Scholar
  21. I. E. Grossmann (ed.), “Global optimization in engineering design,” Kluwer: Dordrecht, 1996b.Google Scholar
  22. I. E. Grossmann, J. A. Caballero, and H. Yeomans, “Advances in mathematical programming for automated design, integration and operation of chemical processes,” Korean J. Chem. Eng. vol. 16, pp. 407–426, 1999.Google Scholar
  23. I. E. Grossmann and M. M. Daichendt, “New trends in optimization-based approaches for process synthesis,” Computers and Chemical Engineering vol. 20, pp. 665–683, 1996.Google Scholar
  24. I. E. Grossmann and S. Lee, “Generalized disjunctive programming: Nonlinear convex hull relaxation,” to appear in Computational Optimization and Applications (2002).Google Scholar
  25. I. E. Grossmann and Z. Kravanja, “Mixed-integer nonlinear programming: A survey of algorithms and applications,” in The IMA Volumes in Mathematics and its Applications, vol. 93: Large-Scale Optimization with Applications, Part II: Optimal Design and Control, Biegler, Coleman, Conn, and Santosa, eds, Springer-Verlag: Berlin, pp. 73–100, 1997.Google Scholar
  26. I. E. Grossmann, J. Quesada, R. Raman and V. Voudouris, “Mixed integer optimization techniques for the design and scheduling of batch processes,” in Batch Processing Systems Engineering, G. V. Reklaitis, A. K. Sunol, D. W. T. Rippin, and O. Hortacsu, eds., Springer-Verlag: Berlin, pp. 451–494, 1996.Google Scholar
  27. O. K. Gupta and V. Ravindran, “Branch and bound experiments in convex nonlinear integer programming,” Management Science vol. 31, no. 12, pp. 1533–1546, 1985.Google Scholar
  28. J. N. Hooker and M. A. Osorio, “Mixed logical linear programming,” Discrete Applied Mathematics vol. 96/97, pp. 395–442, 1999.Google Scholar
  29. J. N. Hooker, Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction, Wiley: New York, 2000.Google Scholar
  30. R. Horst and P. M. Tuy, Global Optimization: Deterministic Approaches, 3rd edn., Springer-Verlag: Berlin, 1996.Google Scholar
  31. E. L. Johnson, G. L. Nemhauser, and M. W. P. Savelsbergh, “Progress in linear programming based branch-andbound algorithms: Exposition,” INFORMS Journal of Computing vol. 12, 2000.Google Scholar
  32. J. Kallrath, “Mixed integer optimization in the chemical process industry: Experience, potential and future,” Trans. I.Chem E. vol. 78, part A, pp. 809–822, 2000.Google Scholar
  33. J. E. Kelley Jr., “The cutting-plane method for solving convex programs,” Journal of SIAM vol. 8, pp. 703–712, 1960.Google Scholar
  34. P. Kesavan and P. I. Barton, “Decomposition algorithms for nonconvex mixed-integer nonlinear programs,” American Institute of Chemical Engineering Symposium Series vol. 96, no. 323, pp. 458–461, 1999.Google Scholar
  35. P. Kesavan and P. I. Barton, “Generalized branch-and-cut framework for mixed-integer nonlinear optimization problems,” Computers and Chem. Engng. vol. 24, pp. 1361–1366, 2000.Google Scholar
  36. G. R. Kocis and I. E. Grossmann, “Relaxation strategy for the structural optimization of process flowsheets,” Ind. Eng. Chem. Res. vol. 26, p. 1869, 1987.Google Scholar
  37. S. Lee and I. E. Grossmann, “New algorithms for nonlinear generalized disjunctive programming,” Computers and Chemical Engineering vol. 24, pp. 2125–2141, 2000.Google Scholar
  38. S. Leyffer, “Deterministic methods for mixed-integer nonlinear programming,” Ph.D. Thesis, Department of Mathematics and Computer Science, University of Dundee, Dundee, 1993.Google Scholar
  39. S. Leyffer, “Integrating SQP and branch and bound for mixed integer noninear programming,” Computational Optimization and Applications vol. 18, pp. 295–309, 2001.Google Scholar
  40. T. L. Magnanti and R. T. Wong, “Acclerated benders decomposition: Algorithm enhancement and model selection criteria,” Operations Research vol. 29, pp. 464–484, 1981.Google Scholar
  41. G. P. McCormick, “Computability of global solutions to factorable nonconvex programs: Part I–Convex underestimating problems,” Mathematical Programming vol. 10, pp. 147–175, 1976.Google Scholar
  42. S. Nabar and L. Schrage, “Modeling and solving nonlinear integer programming problems,” Presented at Annual AIChE Meeting, Chicago, 1991.Google Scholar
  43. G. L. Nemhauser and L. A. Wolsey, Integer and Combinatorial Optimization, Wiley-Interscience: NewYork, 1988.Google Scholar
  44. J. Pinto and I. E. Grossmann, “Assignment and sequencing models for the scheduling of chemical processes,” Annals of Operations Research vol. 81, pp. 433–466, 1998.Google Scholar
  45. R. Pörn and T. Westerlund, “A cutting plane method for minimizing pseudo-convex functions in the mixed-integer case,” Computers and Chemical Engineering vol. 24, pp. 2655–2665, 2000.Google Scholar
  46. I. Quesada and I. E. Grossmann, “An LP/NLP based branch and bound algorithm for convex MINLP optimization problems,” Computers and Chemical Engineering vol. 16, pp. 937–947, 1992.Google Scholar
  47. I. E. Quesada and I. E. Grossmann, “A global optimization algorithm for linear fractional and bilinear programs,” Journal of Global Optimization vol. 6, pp. 39–76, 1995.Google Scholar
  48. R. Raman and I. E. Grossmann, “Relation between MILP modelling and logical inference for chemical process synthesis,” Computers and Chemical Engineering vol. 15, p. 73, 1991.Google Scholar
  49. R. Raman and I. E. Grossmann, “Symbolic integration of logic in mixed integer linear programming techniques for process synthesis,” Computers and Chemical Engineering vol. 17, p. 909, 1993.Google Scholar
  50. R. Raman and I. E. Grossmann, “Modelling and computational techniques for logic based integer programming,” Computers and Chemical Engineering vol. 18, p. 563, 1994.Google Scholar
  51. H. S. Ryoo and N. V. Sahinidis, “Global optimization of nonconvex NLPs and MINLPs with applications in process design,” Computers and Chem. Engng. vol. 19, no. 5, pp. 551–566, 1995.Google Scholar
  52. N. V. Sahinidis, “BARON: A general purpose global optimization software package,” Journal of Global Optimization vol. 8, no. 2, pp. 201–205, 1996.Google Scholar
  53. N. V. Sahinidis and I. E. Grossmann, “Convergence properties of generalized benders decomposition,” Computers and Chemical Engineering vol. 15, p. 481, 1991.Google Scholar
  54. C. A. Schweiger and C. A. Floudas, “Process synthesis, design and control: A mixed integer optimal control framework,” in Proceedings of DYCOPS-5 on Dynamics and Control of Process Systems, 1998, pp. 189–194.Google Scholar
  55. N. Shah, “Single and multisite planning and scheduling: Current status and future challenges,” AIChE Symp. Ser. vol. 94, no. 320, p. 75, 1998.Google Scholar
  56. E. M. B. Smith and C. C. Pantelides, “A symbolic reformulation/spatial branch and bound algorithm for the global optimization of nonconvex MINLPs,” Computers and Chemical Engineering vol. 23, pp. 457–478, 1999.Google Scholar
  57. R. Stubbs and S. Mehrotra, “A branch-and-cut method for 0–1 mixed convex programming,” Mathematical Programming vol. 86, no. 3, pp. 515–532, 1999.Google Scholar
  58. M. Tawarmalani and N. V. Sahinidis, “Global optimization of mixed integer nonlinear programs: A theoretical and computational study,” Mathematical Programming submitted, 2000.Google Scholar
  59. M. Türkay and I. E. Grossmann, “Alogic based outer-approximation algorithm for MINLP optimization of process flowsheets,” Computers and Chemical Enginering vol. 20, pp. 959–978, 1996.Google Scholar
  60. A. Vecchietti and I. E. Grossmann, “LOGMIP: A discrete continuous nonlinear optimizer,” Computers and Chemical Engineering vol. 23, pp. 555–565, 1999.Google Scholar
  61. A. Vecchietti and I. E. Grossmann, “Modeling issues and implementation of language for disjunctive programming,” Computers and Chemical Engineering vol. 24, pp. 2143–2155, 2000.Google Scholar
  62. J. Viswanathan and I. E. Grossmann, “A combined penalty function and outer-approximation method for MINLP optimization,” Comput. Chem. Engng. vol. 14, p. 769, 1990.Google Scholar
  63. T. Westerlund and F. Pettersson, “A cutting plane method for solving convex MINLP problems,” Computers and Chemical Engineering vol. 19, pp. S131–S136, 1995.Google Scholar
  64. H. P. Williams, Mathematical Building in Mathematical Programming, John Wiley: Chichester, 1985.Google Scholar
  65. X. Yuan, S. Zhang, L. Piboleau, and S. Domenech, “Une methode d'optimisation nonlineare en variables mixtes pour la conception de procedes,” RAIRO vol. 22, p. 331, 1988.Google Scholar
  66. J. M. Zamora and I. E. Grossmann, “Abranch and contract algorithm for problems with concave univariate, bilinear and linear fractional terms,” Journal of Gobal Optimization vol. 14, pp. 217–249, 1999.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Ignacio E. Grossmann
    • 1
  1. 1.Department of Chemical EngineeringCarnegie Mellon UniversityPittsburghUSA

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