Abstract
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.
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References
E. Balas, “Disjunctive programming and a hierarchy of relaxations for discrete optimization problems,” SIAM J. Alg. Disc. Meth., vol. 6, pp. 466-486, 1985.
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.
N. Beaumont, “An algorithm for disjunctive programs,” European Journal of Operations Research, vol. 48, pp. 362-371, 1990.
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.
A. Brooke, D. Kendrick, A. Meeraus, and R. Raman, GAMS Language Guide, Release 2.25, Version 92, GAMS Development Corporation, 1997.
S. Ceria and J. Soares, “Convex programming for disjunctive optimization,” Mathematical Programming, vol. 86, pp. 595-614, 1999.
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.
R. Fletcher and S. Leyffer, “Solving mixed nonlinear programs by outer approximation,” Mathematical Programming, vol. 66, pp. 327-349, 1994.
A.M. Geoffrion, “Generalized benders decomposition,” Journal of Optimization Theory and Application, vol. 10, pp. 237-260, 1972.
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.
I.E. Grossmann, “Review of nonlinear mixed-integer and disjunctive programming techniques,” Optimization and Engineering, vol. 3, pp. 227-252, 2002.
O.K. Gupta and V. Ravindran, “Branch and bound experiments in convex nonlinear integer programming,” Management Science, vol. 31, pp. 1533-1546, 1985.
J. Hiriart-Urruty and C. Lemar´echal, Convex Analysis and Minimization Algorithms, vol. 1, Springer-Verlag, 1993.
J.N. Hooker, Logic-Based Methods for Optimization: Combining Optimization and Constraints Satisfaction, Wiley, 2000.
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.
S. Lee and I.E. Grossmann, “New algorithms for nonlinear generalized disjunctive programming,” Computers Chem. Engng., vol. 24, pp. 2125-2141, 2000.
S. Leyffer, “IntegratingSQPand branch-and-bound for mixed integer nonlinear programming,” Computational Optimization and Applications, vol. 18, pp. 295-309, 2001.
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.
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.
R. Raman and I.E. Grossmann, “Modelling and computational techniques for logic based integer programming,” Computers Chem. Engng., vol. 18, pp. 563-578, 1994.
R. Stubbs and S. Mehrotra, “A branch-and-cut method for 0-1 mixed convex programming,” Mathematical Programming, vol. 86, pp. 515-532, 1999.
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.
T.J. Van Roy and L.A. Wolsey, “Solving mixed 0-1 programs by automatic reformulation,” Operations Research, vol. 35, pp. 45-57, 1987.
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.
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.
T. Westerlund and F. Petterson, “An extended cutting plane method for solving convex MINLP problems,” Computers Chem. Engng., vol. 19, pp. S131-S136, 1995.
H.P. Williams, Model Building in Mathematical Programming, John Wiley &; Sons, Inc., 1985.
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.
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Grossmann, I.E., Lee, S. Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation. Computational Optimization and Applications 26, 83–100 (2003). https://doi.org/10.1023/A:1025154322278
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DOI: https://doi.org/10.1023/A:1025154322278