Generalized Disjunctive Programming: Solution Strategies

  • Juan P. Ruiz
  • Jan-H. Jagla
  • Ignacio E. Grossmann
  • Alex Meeraus
  • Aldo Vecchietti
Part of the Applied Optimization book series (APOP, volume 104)


Generalized disjunctive programming (GDP) is an extension of the disjunctive programming paradigm developed by Balas. The GDP formulation involves Boolean and continuous variables that are specified in algebraic constraints, disjunctions and logic propositions, which is an alternative representation to the traditional algebraic mixed-integer programming formulation. GDP has proven to be very useful in representing a wide variety of problems successfully. Even though a wealth of powerful algorithms exist to solve these problems, GDP suffers a lack of mature solver technology. The main goal of this paper is to review the basic concepts and algorithms related to GDP problems and describe how solver technology is being developed. With this in mind after providing a brief review of MINLP optimization, we present an overview of GDP for the case of convex functions emphasizing the quality of continuous relaxations of alternative reformulations that include the big-M and the hull relaxation. We then review disjunctive branch and bound as well as logic-based decomposition methods that circumvent some of the limitations in traditional MINLP optimization. The first implemented GDP solver LogMIP successfully demonstrated that formulating and solving such problems can be done in an algebraic modeling system like GAMS. Recently, LogMIP has been introduced into GAMS’ Extended Mathematical Programming (EMP) framework integrating it much closer into the GAMS system and language and at the same time offering much more flexibility to the user. Since the model is separated from the reformulation chosen and from the solver used to solve the automatically generated model, this setup allows to easily switch methods at no costs and to benefit from advancing solver technology.


Logic Proposition Master Problem Boolean Variable Conjunctive Normal Form Algebraic Constraint 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Abhishek, K., Leyffer, S., Linderoth, J.T.: FilMINT: An Outer-Approximation-Based Solver for Nonlinear Mixed Integer Programs, ANL/MCS-P1374-0906, Argonne National Laboratory (2006)Google Scholar
  2. 2.
    Balas, E.: Disjunctive programming. 5, 3–51 (1979)Google Scholar
  3. 3.
    Balas, E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Alg. Disc. Meth. 6, 466–486 (1985)CrossRefGoogle Scholar
  4. 4.
    Beaumont N.: An algorithm for disjunctive programs. Eur. J. Oper. Res. 48, 362–371 (1991)CrossRefGoogle Scholar
  5. 5.
    Biegler L., Grossmann I.E., Westerberg W.: Systematic methods of chemical process design. Prentice Hall, Englewood Cliffs, NJ, USA (1997)Google Scholar
  6. 6.
    Bonami P., Biegler L.T., Conn A.R., Cornuejols G., Grossmann I.E., Laird C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wächter, A.: An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim. 5, 186–204 (2008)CrossRefGoogle Scholar
  7. 7.
    Borchers, B., Mitchell, J.E.: An improved branch and bound algorithm for mixed integer nonlinear programming. Comput. Oper. Res. 21, 359–367 (1994)CrossRefGoogle Scholar
  8. 8.
    Brooke A., Kendrick, D., Meeraus, A., Raman R.: GAMS, a User’s Guide, GAMS Development Corporation, Washington (1998)Google Scholar
  9. 9.
  10. 10.
    Duran, M.A., Grossmann, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math Program. 36, 307 (1986)CrossRefGoogle Scholar
  11. 11.
    Ferris, M.C., Dirkse, S.P., Jagla, J.-H., Meeraus, A.: An extended mathematical programming framework. Comput. Chem. Eng. 33, 1973–1982 (2009)CrossRefGoogle Scholar
  12. 12.
  13. 13.
    Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer-approximation. Math Program. 66, 327 (1994)CrossRefGoogle Scholar
  14. 14.
  15. 15.
    GAMS Development Corporation, EMP user’s manual, IMA.
  16. 16.
    Geoffrion, A.M.: Generalized Benders decomposition, JOTA, 10, 237–260 (1972)CrossRefGoogle Scholar
  17. 17.
    Grossmann, I.E., Caballero, J.A., Yeomans, H. Advances in mathematical programming for automated design, integration and operation of chemical processes. Korean J. Chem. Eng. 16, 407–426 (1999)CrossRefGoogle Scholar
  18. 18.
    Grossmann, I.E.: Review of non-linear mixed integer and disjunctive programming techiques for process systems engineering. Optim. Eng. 3, 227–252 (2002)CrossRefGoogle Scholar
  19. 19.
    Grossmann, I.E., Lee, S.: Generalized convex disjunctive programming: Nonlinear convex hull relaxation. Comput. Optim. Appl. 26, 83–100 (2003)CrossRefGoogle Scholar
  20. 20.
    Gupta, O.K., Ravindran, V.: Branch and bound experiments in convex nonlinear integer programming. Manag. Sci. 31(12), 1533–1546 (1985)CrossRefGoogle Scholar
  21. 21.
    Hooker, J.N., Osorio, M.A.: Mixed logical-linear programming. Discrete Appl. Math. 96–97, 395–442 (1999)CrossRefGoogle Scholar
  22. 22.
    Hooker, J.N.: Logic-based methods for optimization: Combining optimization and constraint satisfaction. Wiley, NY, USA (2000)CrossRefGoogle Scholar
  23. 23.
    Kallrath, J.: Mixed integer optimization in the chemical process industry: Experience, potential and future, Trans. I.Chem E. 78, 809–822 (2000)Google Scholar
  24. 24.
    Lee, S., Grossmann, I.E.: New algorithms for nonlinear generalized disjunctive programming. Comput. Chem. Eng. 24, 2125–2141 (2000)CrossRefGoogle Scholar
  25. 25.
    Leyffer, S.: Integrating SQP and branch and bound for mixed integer nonlinear programming. Comput. Optim. Appl. 18, 295–309 (2001)CrossRefGoogle Scholar
  26. 26.
    Liberti, L., Mladenovic, M., Nannicini, G.: A good recipe for solving MINLPs. Hybridizing metaheuristics and mathematical programming, Springer, 10 (2009)Google Scholar
  27. 27.
  28. 28.
    Mendez, C.A., Cerda, J., Grossmann, I.E., Harjunkoski I., Fahl, M.: State-of-the-art review of optimization methods for short-term scheduling of batch processes. Comput. Chem. Eng. 30, 913 (2006)CrossRefGoogle Scholar
  29. 29.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization, Wiley, New York (1988)Google Scholar
  30. 30.
    Quesada, I., Grossmann, I.E.: An LP/NLP based branch and bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16, 937–947 (1992)CrossRefGoogle Scholar
  31. 31.
    Raman, R., Grossmann, I.E.: Modeling and computational techniques for logic-based integer programming. Comput. Chem. Eng. 18, 563 (1994)CrossRefGoogle Scholar
  32. 32.
    Grossmann, I.E., Ruiz, J.P.: Generalized Disjunctive Programming: a framework for formulation and alternative MINLP optimization. In: Lee, J., Leyffer, S. (Eds.), Mixed Integer Nonlinear Programming. Series: The IMA Volumes in Mathematics and Its Applications, vol. 154, pp. 93–115. Springer, New York (2012)CrossRefGoogle Scholar
  33. 33.
    Ruiz, J.P., Grossmann, I.E.: A hierarchy of relaxations for convex generalized disjunctive programs. European Journal of Operational Research 218, 38–47 (2012)CrossRefGoogle Scholar
  34. 34.
    Sahinidis, N.V.: BARON: A general purpose global optimization software package. J. Global Optim. 8(2), 201–205 (1996)CrossRefGoogle Scholar
  35. 35.
    Sawaya, N.: Thesis: Reformulations, relaxations and cutting planes for generalized disjunctive programming. Carnegie Mellon University, Pittsburgh, PA (2006)Google Scholar
  36. 36.
  37. 37.
    Stubbs, R., Mehrotra, S.: A Branch-and-cut method for 0-1 mixed convex programming. Math Program. 86(3), 515–532 (1999)CrossRefGoogle Scholar
  38. 38.
    Turkay, M., Grossmann, I.E.: A Logic-based outer-approximation algorithm for MINLP optimization of process flowsheets. Comput. Chem. Eng. 20, 959–978 (1996)CrossRefGoogle Scholar
  39. 39.
    Vecchietti, A., Grossmann, I.E.: Logmip: A disjunctive 01 non-linear optimizer for process system models. Comput. Chem. Eng. 23(4), 555–565 (1999)CrossRefGoogle Scholar
  40. 40.
    Vecchietti, A., Lee, S., Grossmann, I.E.: Modeling of discrete/continuous optimization problems: Characterization and formulation of disjunctions and their relaxations. Comput. Chem. Eng. 27, 433–448 (2003)CrossRefGoogle Scholar
  41. 41.
    Vecchietti, A., Grossmann, I.E.: LOGMIP: A discrete continuous nonlinear optimizer. Comput. Chem. Eng. 23, 555–565 (2003)Google Scholar
  42. 42.
    Viswanathan and Grossmann I.E.: A combined penalty function and outer-approximation method for MINLP optimization. Comput. Chem. Eng. 14, 769–782 (1990)Google Scholar
  43. 43.
    Westerlund, T., Pettersson, F.: A Cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, S131–S136 (1995)CrossRefGoogle Scholar
  44. 44.
    Westerlund, T., Pörn, R.: Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3, 253–280 (2002)CrossRefGoogle Scholar
  45. 45.
    Williams, H.P.: Mathematical building in mathematical programming. Wiley, New York (1985)Google Scholar
  46. 46.
    Yuan, X., Zhang, S., Piboleau, L., Domenech, S.: Une methode d’optimisation nonlineare en variables mixtes pour la conception de procedes. RAIRO, 22, 331 (1988)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Juan P. Ruiz
    • 1
  • Jan-H. Jagla
    • 2
  • Ignacio E. Grossmann
    • 1
  • Alex Meeraus
    • 3
  • Aldo Vecchietti
    • 4
  1. 1.Carnegie Mellon UniversityPittsburghUSA
  2. 2.GAMS Software GmbHBraunschweigGermany
  3. 3.GAMS Development CorporationWashingtonUSA
  4. 4.Universidad Tecnológica NacionalSanta FeUSA

Personalised recommendations