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A Hierarchy of Relaxations Leading to the Convex Hull Representation for General Discrete Optimization Problems

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  • Warren Adams
  • Hanif Sherali

Abstract

We consider linear mixed-integer programs where a subset of the variables are restricted to take on a finite number of general discrete values. For this class of problems, we develop a reformulation-linearization technique (RLT) to generate a hierarchy of linear programming relaxations that spans the spectrum from the continuous relaxation to the convex hull representation. This process involves a reformulation phase in which suitable products using a defined set of Lagrange interpolating polynomials (LIPs) are constructed, accompanied by the application of an identity that generalizes x(1−x) for the special case of a binary variable x. This is followed by a linearization phase that is based on variable substitutions. The constructs and arguments are distinct from those for the mixed 0-1 RLT, yet they encompass these earlier results. We illustrate the approach through some examples, emphasizing the polyhedral structure afforded by the linearized LIPs. We also consider polynomial mixed-integer programs, exploitation of structure, and conditional-logic enhancements, and provide insight into relationships with a special-structure RLT implementation. Copyright Springer Science + Business Media, Inc. 2005

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  • Warren Adams & Hanif Sherali, 2005. "A Hierarchy of Relaxations Leading to the Convex Hull Representation for General Discrete Optimization Problems," Annals of Operations Research, Springer, vol. 140(1), pages 21-47, November.
    • Handle: RePEc:spr:annopr:v:140:y:2005:i:1:p:21-47:10.1007/s10479-005-3966-4
    DOI: 10.1007/s10479-005-3966-4
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    Cited by:

    1. Binyuan Chen & Simge Küçükyavuz & Suvrajeet Sen, 2011. "Finite Disjunctive Programming Characterizations for General Mixed-Integer Linear Programs," Operations Research, INFORMS, vol. 59(1), pages 202-210, February.
    2. Lucas A. Waddell & Jerry L. Phillips & Tianzhu Liu & Swarup Dhar, 2023. "An LP-based characterization of solvable QAP instances with chess-board and graded structures," Journal of Combinatorial Optimization, Springer, vol. 45(5), pages 1-23, July.
    3. Martin Ballerstein & Dennis Michaels, 2014. "Extended formulations for convex envelopes," Journal of Global Optimization, Springer, vol. 60(2), pages 217-238, October.
    4. Scott J. Davis & Shatiel B. Edwards & Gerald E. Teper & David G. Bassett & Michael J. McCarthy & Scott C. Johnson & Craig R. Lawton & Matthew J. Hoffman & Liliana Shelton & Stephen M. Henry & Darryl J, 2016. "Maximizing the U.S. Army’s Future Contribution to Global Security Using the Capability Portfolio Analysis Tool (CPAT)," Interfaces, INFORMS, vol. 46(1), pages 91-108, February.
    5. Guanglei Wang & Hassan Hijazi, 2018. "Mathematical programming methods for microgrid design and operations: a survey on deterministic and stochastic approaches," Computational Optimization and Applications, Springer, vol. 71(2), pages 553-608, November.

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