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A computationally useful algebraic representation of nonlinear disjunctive convex sets using the perspective function

Author

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  • Kevin C. Furman

    (ExxonMobil Upstream Research Company)

  • Nicolas W. Sawaya

    (ExxonMobil Upstream Integrated Solutions Company)

  • Ignacio E. Grossmann

    (Carnegie Mellon University)

Abstract

Nonlinear disjunctive convex sets arise naturally in the formulation or solution methods of many discrete–continuous optimization problems. Often, a tight algebraic representation of the disjunctive convex set is sought, with the tightest such representation involving the characterization of the convex hull of the disjunctive convex set. In the most general case, this can be explicitly expressed through the use of the perspective function in higher dimensional space—the so-called extended formulation of the convex hull of a disjunctive convex set. However, there are a number of challenges in using this characterization in computation which prevents its wide-spread use, including issues that arise because of the functional form of the perspective function. In this paper, we propose an explicit algebraic representation of a fairly large class of nonlinear disjunctive convex sets using the perspective function that addresses this latter computational challenge. This explicit representation can be used to generate (tighter) algebraic reformulations for a variety of different problems containing disjunctive convex sets, and we report illustrative computational results using this representation for several nonlinear disjunctive problems.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:coopap:v:76:y:2020:i:2:d:10.1007_s10589-020-00176-0
    DOI: 10.1007/s10589-020-00176-0
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    References listed on IDEAS

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    1. 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.
    2. Antonio Frangioni & Claudio Gentile & Enrico Grande & Andrea Pacifici, 2011. "Projected Perspective Reformulations with Applications in Design Problems," Operations Research, INFORMS, vol. 59(5), pages 1225-1232, October.
    3. 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.
    4. Samir Elhedhli, 2006. "Service System Design with Immobile Servers, Stochastic Demand, and Congestion," Manufacturing & Service Operations Management, INFORMS, vol. 8(1), pages 92-97, December.
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