Generalized Disjunctive Programming: A Framework for Formulation and Alternative Algorithms for MINLP Optimization

Conference paper
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 154)

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. 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. We next consider the case of linear GDP problems to show how a hierarchy of relaxations can be developed by performing sequential intersection of disjunctions. Finally, for the case when the GDP problem involves nonconvex functions, we propose a scheme for tightening the lower bounds for obtaining the global optimum using a combined disjunctive and spatial branch and bound search. We illustrate the application of the theoretical concepts and algorithms on several engineering and OR problems.

Key words

Disjunctive programming Mixed-integer nonlinear programming global optimization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abhishek K., Leyffer S., and 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.CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Beaumont N., An Algorithm for Disjunctive Programs, European Journal of Operations Research, 48, 362–371, 1991.CrossRefGoogle Scholar
  5. 5.
    Belotti P., Lee J., Liberti L., Margot F., and W¨achter A., Branching and bounds tightening techniques for non-convex MINLP, Optimization Methods and Software, 24:4, 597–634, 2009.Google Scholar
  6. 6.
    Benders J.F., Partitioning procedures for solving mixed-variables programming problems, Numer.Math., 4, 238–252, 1962.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Borchers B. and Mitchell J.E., An Improved Branch and Bound Algorithm for Mixed Integer Nonlinear Programming, Computers and Operations Research, 21, 359–367, 1994.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Bonami P., Biegler L.T., Conn A.R., Cornuejols G., Grossmann I.E., Laird C.D., Lee J., Lodi A., Margot F., Sawaya N., and W¨achter A., An algorithmic framework for convex mixed integer nonlinear programs, Discrete Optimization, 5, 186–204, 2008.Google Scholar
  9. 9.
    Brooke A., Kendrick D., Meeraus A., and Raman R., GAMS, a User’s Guide, GAMS Development Corporation, Washington, 1998.Google Scholar
  10. 10.
    Duran M.A. and Grossmann I.E., An Outer-Approximation Algorithm for a Class of Mixed-integer Nonlinear Programs, Math Programming, 36, p. 307, 1986.CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Fletcher R. and Leyffer S., Solving Mixed Integer Nonlinear Programs by Outer-Approximation, Math Programming, 66, p. 327, 1994.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Galan B. and Grossmann I.E., Optimal Design of Distributed Wastewater Treatment Networks, Ind. Eng. Chem. Res., 37, 4036–4048, 1998.CrossRefGoogle Scholar
  13. 13.
    Geoffrion A.M., Generalized Benders decomposition, JOTA, 10, 237–260, 1972.CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Grossmann I.E., Review of Non-Linear Mixed Integer and Disjunctive Programming Techiques for Process Systems Engineering, Optimization and Engineering, 3, 227–252, 2002.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Grossmann I.E., Caballero J.A., and 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
  16. 16.
    Grossmann I.E. and Lee S., Generalized Convex Disjunctive Programming: Nonlinear Convex Hull Relaxation, Computational Optimization and Applications, 26, 83–100, 2003.CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Gupta O.K. and Ravindran V., Branch and Bound Experiments in Convex Non-linear Integer Programming, Management Science, 31:12, 1533–1546, 1985.Google Scholar
  18. 18.
    Hooker J.N. and Osorio M.A., Mixed logical-linear programming, Discrete Applied Mathematics, 96–97, 395–442, 1999.CrossRefMathSciNetGoogle Scholar
  19. 19.
    Hooker J.N., Logic-Based Methods for Optimization: Combining Optimization and Constraint Satisfaction, Wiley, 2000.Google Scholar
  20. 20.
    Horst R. and Tuy H., Global Optimization deterministic approaches, 3rd Ed, Springer-Verlag, 1996.Google Scholar
  21. 21.
    Kallrath J., Mixed Integer Optimization in the Chemical Process Industry: Experience, Potential and Future, Trans. I. Chem E., 78, 809–822, 2000.Google Scholar
  22. 22.
    Lee S. and Grossmann I.E., New Algorithms for Nonlinear Generalized Disjunctive Programming, Computers and Chemical Engineering, 24, 2125–2141, 2000.CrossRefGoogle Scholar
  23. 23.
    Lee S. and Grossmann I.E., Global optimization of nonlinear generalized disjunctive programming with bilinear inequality constraints: application to process networks, Computers and Chemical Engineering, 27, 1557–1575, 2003.CrossRefGoogle Scholar
  24. 24.
    Leyffer S., Integrating SQP and Branch and Bound for Mixed Integer Nonlinear Programming, Computational Optimization and Applications, 18, 295–309, 2001.CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Liberti L., Mladenovic M., and Nannicini G., A good recipe for solving MINLPs, Hybridizing metaheuristics and mathematical programming, Springer, 10, 2009.Google Scholar
  26. 26.
    Lindo Systems Inc, LindoGLOBAL SolverGoogle Scholar
  27. 27.
    Mendez C.A., Cerda J., Grossmann I.E., Harjunkoski I., and Fahl M., State-of-the-art Review of Optimization Methods for Short-Term Scheduling of Batch Processes, Comput. Chem. Eng., 30, p. 913, 2006.CrossRefGoogle Scholar
  28. 28.
    Nemhauser G.L. and Wolsey L.A., Integer and Combinatorial Optimization, Wiley-Interscience, 1988.Google Scholar
  29. 29.
    Quesada I. and Grossmann I.E., An LP/NLP Based Branch and Bound Algorithm for Convex MINLP Optimization Problems, Computers and Chemical Engineering, 16, 937–947, 1992.CrossRefGoogle Scholar
  30. 30.
    Quesada I. and Grossmann I.E., Global optimization of bilinear process networks with multicomponent flows, Computers and Chemical Engineering, 19:12, 1219–1242, 1995.Google Scholar
  31. 31.
    Raman R. and Grossmann I.E., Relation Between MILP Modelling and Logical Inference for Chemical Process Synthesis, Computers and Chemical Engineer- ing, 15, 73, 1991.CrossRefGoogle Scholar
  32. 32.
    Raman R. and Grossmann I.E., Modelling and Computational Techniques for Logic-Based Integer Programming, Computers and Chemical Engineering, 18, p. 563, 1994.CrossRefGoogle Scholar
  33. 33.
    Ruiz J.P. and Grossmann I.E., Strengthening the lower bounds for bilinear and concave GDP problems, Computers and Chemical Engineering, 34:3, 914–930, 2010.Google Scholar
  34. 34.
    Sahinidis N.V., BARON: A General Purpose Global Optimization Software Package, Journal of Global Optimization, 8:2, 201–205, 1996.Google Scholar
  35. 35.
    Sawaya N. and Grossmann I.E., A cutting plane method for solving linear generalized disjunctive programming problems, Computers and Chemical Engineering, 20:9, 1891–1913, 2005.Google Scholar
  36. 36.
    Sawaya N., Thesis: Reformulations, relaxations and cutting planes for generalized disjunctive programming, Carnegie Mellon University, 2006.Google Scholar
  37. 37.
    Schweiger C.A. and Floudas C.A., Process Synthesis, Design and Control: A Mixed Integer Optimal Control Framework, Proceedings of DYCOPS-5 on Dynamics and Control of Process Systems, 189–194, 1998.Google Scholar
  38. 38.
    Stubbs R. and Mehrotra S., A Branch-and-Cut Method for 0–1 Mixed Convex Programming, Math Programming, 86:3, 515–532, 1999.CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Tawarmalani M. and Sahinidis N., Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming, Kluwer Academic Publishers, 2002.Google Scholar
  40. 40.
    Turkay M. and Grossmann I.E., A Logic-Based Outer-Approximation Algorithm for MINLP Optimization of Process Flowsheets, Computers and Chemical Enginering, 20, 959–978, 1996.CrossRefGoogle Scholar
  41. 41.
    Vecchietti A., Lee S., and Grossmann, I.E., Modeling of discrete/continuous optimization problems: characterization and formulation of disjunctions and their relaxations, Computers and Chemical Engineering, 27,433–448, 2003.CrossRefGoogle Scholar
  42. 42.
    Vecchietti A. and Grossmann I.E., LOGMIP: A Discrete Continuous Nonlinear Optimizer, Computers and Chemical Engineering, 23, 555–565, 2003.CrossRefGoogle Scholar
  43. 43.
    Viswanathan and Grossmann I.E., A combined penalty function and outer approximation method for MINLP optimization, Computers and Chemical Engineering, 14, 769–782, 1990.Google Scholar
  44. 44.
    Westerlund T. and Pettersson F., A Cutting Plane Method for Solving Convex MINLP Problems, Computers and Chemical Engineering, 19, S131–S136, 1995.CrossRefGoogle Scholar
  45. 45.
    Westerlund T. and P¨orn R., Solving Pseudo-Convex Mixed Integer Optimization Problems by Cutting Plane Techniques, Optimization and Engineering, 3, 253–280, 2002.Google Scholar
  46. 46.
    Williams H.P., Mathematical Building in Mathematical Programming, John Wiley, 1985.Google Scholar
  47. 47.
    Yuan, X., Zhang S., Piboleau L., and Domenech S., Une Methode d’optimisation Nonlineare en Variables Mixtes pour la Conception de Procedes, RAIRO, 22, 331, 1988.MATHGoogle Scholar
  48. 48.
    Zamora J.M. and Grossmann I.E., A branch and bound algorithm for problems with concave univariate, bilinear and linear fractional terms, 14:3, 217–249,1999.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Carnegie Mellon UniversityPittsburghUSA

Personalised recommendations