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Pseudo basic steps: bound improvement guarantees from Lagrangian decomposition in convex disjunctive programming

Author

Listed:
  • Dimitri J. Papageorgiou

    (ExxonMobil Research and Engineering Company)

  • Francisco Trespalacios

    (ExxonMobil Research and Engineering Company)

Abstract

An elementary, but fundamental, operation in disjunctive programming is a basic step, which is the intersection of two disjunctions to form a new disjunction. Basic steps bring a disjunctive set in regular form closer to its disjunctive normal form and, in turn, produce relaxations that are at least as tight. An open question is: What are guaranteed bounds on the improvement from a basic step? In this paper, using properties of a convex disjunctive program’s hull reformulation and multipliers from Lagrangian decomposition, we introduce an operation called a pseudo basic step and use it to provide provable bounds on this improvement along with techniques to exploit this information when solving a disjunctive program as a convex MINLP. Numerical examples illustrate the practical benefits of these bounds. In particular, on a set of K-means clustering instances, we make significant bound improvements relative to state-of-the-art commercial mixed-integer programming solvers.

Suggested Citation

  • Dimitri J. Papageorgiou & Francisco Trespalacios, 2018. "Pseudo basic steps: bound improvement guarantees from Lagrangian decomposition in convex disjunctive programming," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 6(1), pages 55-83, March.
  • Handle: RePEc:spr:eurjco:v:6:y:2018:i:1:d:10.1007_s13675-017-0088-0
    DOI: 10.1007/s13675-017-0088-0
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    References listed on IDEAS

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    1. Trespalacios, Francisco & Grossmann, Ignacio E., 2016. "Lagrangean relaxation of the hull-reformulation of linear generalized disjunctive programs and its use in disjunctive branch and bound," European Journal of Operational Research, Elsevier, vol. 253(2), pages 314-327.
    2. Sawaya, Nicolas & Grossmann, Ignacio, 2012. "A hierarchy of relaxations for linear generalized disjunctive programming," European Journal of Operational Research, Elsevier, vol. 216(1), pages 70-82.
    3. Ruiz, Juan P. & Grossmann, Ignacio E., 2012. "A hierarchy of relaxations for nonlinear convex generalized disjunctive programming," European Journal of Operational Research, Elsevier, vol. 218(1), pages 38-47.
    4. Monique Guignard, 2003. "Lagrangean relaxation," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 11(2), pages 151-200, December.
    5. Pietro Belotti & Pierre Bonami & Matteo Fischetti & Andrea Lodi & Michele Monaci & Amaya Nogales-Gómez & Domenico Salvagnin, 2016. "On handling indicator constraints in mixed integer programming," Computational Optimization and Applications, Springer, vol. 65(3), pages 545-566, December.
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