Computational Optimization and Applications

, Volume 26, Issue 1, pp 83–100 | Cite as

Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation

  • Ignacio E. Grossmann
  • Sangbum Lee


Generalized Disjunctive Programming (GDP) has been introduced recently as an alternative to mixed-integer programming for representing discrete/continuous optimization problems. The basic idea of GDP consists of representing these problems in terms of sets of disjunctions in the continuous space, and logic propositions in terms of Boolean variables. In this paper we consider GDP problems involving convex nonlinear inequalities in the disjunctions. Based on the work by Stubbs and Mehrotra [21] and Ceria and Soares [6], we propose a convex nonlinear relaxation of the nonlinear convex GDP problem that relies on the convex hull of each of the disjunctions that is obtained by variable disaggregation and reformulation of the inequalities. The proposed nonlinear relaxation is used to formulate the GDP problem as a Mixed-Integer Nonlinear Programming (MINLP) problem that is shown to be tighter than the conventional “big-M” formulation. A disjunctive branch and bound method is also presented, and numerical results are given for a set of test problems.

disjunctive programming convex programming mixed integer nonlinear programming convex hull 


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  1. 1.
    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
  2. 2.
    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
  3. 3.
    N. Beaumont, “An algorithm for disjunctive programs,” European Journal of Operations Research, vol. 48, pp. 362-371, 1990.Google Scholar
  4. 4.
    B. Borchers and J.E. Mitchell, “An improved branch and bound algorithm for mixed integer nonlinear programming,” Computers and Operations Research, vol. 21, pp. 395-367, 1994.Google Scholar
  5. 5.
    A. Brooke, D. Kendrick, A. Meeraus, and R. Raman, GAMS Language Guide, Release 2.25, Version 92, GAMS Development Corporation, 1997.Google Scholar
  6. 6.
    S. Ceria and J. Soares, “Convex programming for disjunctive optimization,” Mathematical Programming, vol. 86, pp. 595-614, 1999.Google Scholar
  7. 7.
    M.A. Duran and I.E. Grossmann, “An outer-approximation algorithm for a class of mixed-integer nonlinear programs,” Mathematical Programming, vol. 36, pp. 307-339, 1986.Google Scholar
  8. 8.
    R. Fletcher and S. Leyffer, “Solving mixed nonlinear programs by outer approximation,” Mathematical Programming, vol. 66, pp. 327-349, 1994.Google Scholar
  9. 9.
    A.M. Geoffrion, “Generalized benders decomposition,” Journal of Optimization Theory and Application, vol. 10, pp. 237-260, 1972.Google Scholar
  10. 10.
    I.E. Grossmann and Z. Kravanja, “Mixed-integer nonlinear programming: A survey of algorithms and applications,” Large-Scale Optimization with Applications, Part II: Optimal Design and Control, L.T. Biegler et al. (Eds.), Springer-Verlag, 1997, pp. 73-100.Google Scholar
  11. 11.
    I.E. Grossmann, “Review of nonlinear mixed-integer and disjunctive programming techniques,” Optimization and Engineering, vol. 3, pp. 227-252, 2002.Google Scholar
  12. 12.
    O.K. Gupta and V. Ravindran, “Branch and bound experiments in convex nonlinear integer programming,” Management Science, vol. 31, pp. 1533-1546, 1985.Google Scholar
  13. 13.
    J. Hiriart-Urruty and C. Lemar´echal, Convex Analysis and Minimization Algorithms, vol. 1, Springer-Verlag, 1993.Google Scholar
  14. 14.
    J.N. Hooker, Logic-Based Methods for Optimization: Combining Optimization and Constraints Satisfaction, Wiley, 2000.Google Scholar
  15. 15.
    E.L. Johnson, G.L. Nemhauser and M.W.P. Savelsbergh, “Progress in linear programming based branch-andbound algorithms: An exposition,” INFORMS Journal on Computing, vol. 12, pp. 2-23, 2000.Google Scholar
  16. 16.
    S. Lee and I.E. Grossmann, “New algorithms for nonlinear generalized disjunctive programming,” Computers Chem. Engng., vol. 24, pp. 2125-2141, 2000.Google Scholar
  17. 17.
    S. Leyffer, “IntegratingSQPand branch-and-bound for mixed integer nonlinear programming,” Computational Optimization and Applications, vol. 18, pp. 295-309, 2001.Google Scholar
  18. 18.
    I. Quesada and I.E. Grossmann, “An LP/NLP based branch and bound algorithm for convex MINLP optimization problems,” Computers Chem. Engng., vol. 16, pp. 937-947, 1992.Google Scholar
  19. 19.
    R. Raman and I.E. Grossmann, “Symbolic integration of logic in MILP branch and bound methods for the synthesis of process networks,” Annals of Operations Research, vol. 42, pp. 169-191, 1993.Google Scholar
  20. 20.
    R. Raman and I.E. Grossmann, “Modelling and computational techniques for logic based integer programming,” Computers Chem. Engng., vol. 18, pp. 563-578, 1994.Google Scholar
  21. 21.
    R. Stubbs and S. Mehrotra, “A branch-and-cut method for 0-1 mixed convex programming,” Mathematical Programming, vol. 86, pp. 515-532, 1999.Google Scholar
  22. 22.
    M. T¨urkay and I.E. Grossmann, “Logic-based MINLP algorithms for the optimal synthesis of process networks,” Computers Chem. Engng., vol. 20, pp. 959-978, 1996.Google Scholar
  23. 23.
    T.J. Van Roy and L.A. Wolsey, “Solving mixed 0-1 programs by automatic reformulation,” Operations Research, vol. 35, pp. 45-57, 1987.Google Scholar
  24. 24.
    A. Vecchietti and I.E. Grossmann, “LOGMIP: A disjunctive 0-1 nonlinear optimizer for process systems models,” Computers Chem. Engng., vol. 23, pp. 555-565, 1999.Google Scholar
  25. 25.
    J. Viswanathan and I.E. Grossmann, “A combined penalty function and outer-approximation method for MINLP optimization,” Computers Chem. Engng., vol. 14, pp. 769-782, 1990.Google Scholar
  26. 26.
    T. Westerlund and F. Petterson, “An extended cutting plane method for solving convex MINLP problems,” Computers Chem. Engng., vol. 19, pp. S131-S136, 1995.Google Scholar
  27. 27.
    H.P. Williams, Model Building in Mathematical Programming, John Wiley &; Sons, Inc., 1985.Google Scholar
  28. 28.
    X. Yuan, S. Zhang, L. Piboleau, and S. Domenech, “Une methode d'optimization nonlineare en variables mixtes pour la conception de porcedes,” Rairo Recherche Operationnele, vol. 22, p. 331, 1988.Google Scholar

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© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Ignacio E. Grossmann
    • 1
  • Sangbum Lee
    • 1
  1. 1.Department of Chemical EngineeringCarnegie Mellon UniversityPittsburgh

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