Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
- 2.
For comparison, GAMS systems up to 23.6 contain the original LogMIP implementation and are still available for download at http://www.gams.com/download/download\_old.htm
- 3.
ConvexHull, big = 1e4, eps = 1e-4.
- 4.
References
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)
Balas, E.: Disjunctive programming. 5, 3–51 (1979)
Balas, E.: Disjunctive programming and a hierarchy of relaxations for discrete optimization problems. SIAM J. Alg. Disc. Meth. 6, 466–486 (1985)
Beaumont N.: An algorithm for disjunctive programs. Eur. J. Oper. Res. 48, 362–371 (1991)
Biegler L., Grossmann I.E., Westerberg W.: Systematic methods of chemical process design. Prentice Hall, Englewood Cliffs, NJ, USA (1997)
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)
Borchers, B., Mitchell, J.E.: An improved branch and bound algorithm for mixed integer nonlinear programming. Comput. Oper. Res. 21, 359–367 (1994)
Brooke A., Kendrick, D., Meeraus, A., Raman R.: GAMS, a User’s Guide, GAMS Development Corporation, Washington (1998)
Duran, M.A., Grossmann, I.E.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math Program. 36, 307 (1986)
Ferris, M.C., Dirkse, S.P., Jagla, J.-H., Meeraus, A.: An extended mathematical programming framework. Comput. Chem. Eng. 33, 1973–1982 (2009)
Fletcher, R., Leyffer, S.: Solving mixed integer nonlinear programs by outer-approximation. Math Program. 66, 327 (1994)
GAMS Development Corporation, EMP user’s manual, IMA. www.gams.com/solvers/emp.pdf
Geoffrion, A.M.: Generalized Benders decomposition, JOTA, 10, 237–260 (1972)
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)
Grossmann, I.E.: Review of non-linear mixed integer and disjunctive programming techiques for process systems engineering. Optim. Eng. 3, 227–252 (2002)
Grossmann, I.E., Lee, S.: Generalized convex disjunctive programming: Nonlinear convex hull relaxation. Comput. Optim. Appl. 26, 83–100 (2003)
Gupta, O.K., Ravindran, V.: Branch and bound experiments in convex nonlinear integer programming. Manag. Sci. 31(12), 1533–1546 (1985)
Hooker, J.N., Osorio, M.A.: Mixed logical-linear programming. Discrete Appl. Math. 96–97, 395–442 (1999)
Hooker, J.N.: Logic-based methods for optimization: Combining optimization and constraint satisfaction. Wiley, NY, USA (2000)
Kallrath, J.: Mixed integer optimization in the chemical process industry: Experience, potential and future, Trans. I.Chem E. 78, 809–822 (2000)
Lee, S., Grossmann, I.E.: New algorithms for nonlinear generalized disjunctive programming. Comput. Chem. Eng. 24, 2125–2141 (2000)
Leyffer, S.: Integrating SQP and branch and bound for mixed integer nonlinear programming. Comput. Optim. Appl. 18, 295–309 (2001)
Liberti, L., Mladenovic, M., Nannicini, G.: A good recipe for solving MINLPs. Hybridizing metaheuristics and mathematical programming, Springer, 10 (2009)
LindoGLOBAL Solver, http://www.gams.com/dd/docs/solvers/lindoglobal.pdf
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)
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization, Wiley, New York (1988)
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)
Raman, R., Grossmann, I.E.: Modeling and computational techniques for logic-based integer programming. Comput. Chem. Eng. 18, 563 (1994)
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)
Ruiz, J.P., Grossmann, I.E.: A hierarchy of relaxations for convex generalized disjunctive programs. European Journal of Operational Research 218, 38–47 (2012)
Sahinidis, N.V.: BARON: A general purpose global optimization software package. J. Global Optim. 8(2), 201–205 (1996)
Sawaya, N.: Thesis: Reformulations, relaxations and cutting planes for generalized disjunctive programming. Carnegie Mellon University, Pittsburgh, PA (2006)
Stubbs, R., Mehrotra, S.: A Branch-and-cut method for 0-1 mixed convex programming. Math Program. 86(3), 515–532 (1999)
Turkay, M., Grossmann, I.E.: A Logic-based outer-approximation algorithm for MINLP optimization of process flowsheets. Comput. Chem. Eng. 20, 959–978 (1996)
Vecchietti, A., Grossmann, I.E.: Logmip: A disjunctive 01 non-linear optimizer for process system models. Comput. Chem. Eng. 23(4), 555–565 (1999)
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)
Vecchietti, A., Grossmann, I.E.: LOGMIP: A discrete continuous nonlinear optimizer. Comput. Chem. Eng. 23, 555–565 (2003)
Viswanathan and Grossmann I.E.: A combined penalty function and outer-approximation method for MINLP optimization. Comput. Chem. Eng. 14, 769–782 (1990)
Westerlund, T., Pettersson, F.: A Cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19, S131–S136 (1995)
Westerlund, T., Pörn, R.: Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3, 253–280 (2002)
Williams, H.P.: Mathematical building in mathematical programming. Wiley, New York (1985)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Ruiz, J.P., Jagla, JH., Grossmann, I.E., Meeraus, A., Vecchietti, A. (2012). Generalized Disjunctive Programming: Solution Strategies. In: Kallrath, J. (eds) Algebraic Modeling Systems. Applied Optimization, vol 104. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23592-4_4
Download citation
DOI: https://doi.org/10.1007/978-3-642-23592-4_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-23591-7
Online ISBN: 978-3-642-23592-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)