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Logic-Based Modeling and Solution of Nonlinear Discrete/Continuous Optimization Problems

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  • Sangbum Lee
  • Ignacio Grossmann

Abstract

This paper presents a review of advances in the mathematical programming approach to discrete/continuous optimization problems. We first present a brief review of MILP and MINLP for the case when these problems are modeled with algebraic equations and inequalities. Since algebraic representations have some limitations such as difficulty of formulation and numerical singularities for the nonlinear case, we consider logic-based modeling as an alternative approach, particularly Generalized Disjunctive Programming (GDP), which the authors have extensively investigated over the last few years. Solution strategies for GDP models are reviewed, including the continuous relaxation of the disjunctive constraints. Also, we briefly review a hybrid model that integrates disjunctive programming and mixed-integer programming. Finally, the global optimization of nonconvex GDP problems is discussed through a two-level branch and bound procedure. Copyright Springer Science + Business Media, Inc. 2005

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  • Sangbum Lee & Ignacio Grossmann, 2005. "Logic-Based Modeling and Solution of Nonlinear Discrete/Continuous Optimization Problems," Annals of Operations Research, Springer, vol. 139(1), pages 267-288, October.
    • Handle: RePEc:spr:annopr:v:139:y:2005:i:1:p:267-288:10.1007/s10479-005-3451-0
    DOI: 10.1007/s10479-005-3451-0
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    1. Harlan Crowder & Ellis L. Johnson & Manfred Padberg, 1983. "Solving Large-Scale Zero-One Linear Programming Problems," Operations Research, INFORMS, vol. 31(5), pages 803-834, October.
    2. Ellis L. Johnson & George L. Nemhauser & Martin W.P. Savelsbergh, 2000. "Progress in Linear Programming-Based Algorithms for Integer Programming: An Exposition," INFORMS Journal on Computing, INFORMS, vol. 12(1), pages 2-23, February.
    3. Egon Balas, 1965. "An Additive Algorithm for Solving Linear Programs with Zero-One Variables," Operations Research, INFORMS, vol. 13(4), pages 517-546, August.
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    Cited by:

    1. Francisco Trespalacios & Ignacio E. Grossmann, 2015. "Algorithmic Approach for Improved Mixed-Integer Reformulations of Convex Generalized Disjunctive Programs," INFORMS Journal on Computing, INFORMS, vol. 27(1), pages 59-74, February.
    2. Francisco Trespalacios & Ignacio E. Grossmann, 2016. "Cutting Plane Algorithm for Convex Generalized Disjunctive Programs," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 209-222, May.
    3. Kevin C. Furman & Nicolas W. Sawaya & Ignacio E. Grossmann, 2020. "A computationally useful algebraic representation of nonlinear disjunctive convex sets using the perspective function," Computational Optimization and Applications, Springer, vol. 76(2), pages 589-614, June.
    4. Caballero, José A. & Navarro, Miguel A. & Ruiz-Femenia, Rubén & Grossmann, Ignacio E., 2014. "Integration of different models in the design of chemical processes: Application to the design of a power plant," Applied Energy, Elsevier, vol. 124(C), pages 256-273.

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