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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

Suggested Citation

  • 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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    1. Lawrence J. Watters, 1967. "Letter to the Editor—Reduction of Integer Polynomial Programming Problems to Zero-One Linear Programming Problems," Operations Research, INFORMS, vol. 15(6), pages 1171-1174, December.
    2. Fred Glover & Eugene Woolsey, 1973. "Further Reduction of Zero-One Polynomial Programming Problems to Zero-One linear Programming Problems," Operations Research, INFORMS, vol. 21(1), pages 156-161, February.
    3. Christopher E. Nugent & Thomas E. Vollmann & John Ruml, 1968. "An Experimental Comparison of Techniques for the Assignment of Facilities to Locations," Operations Research, INFORMS, vol. 16(1), pages 150-173, February.
    4. Warren P. Adams & Hanif D. Sherali, 1986. "A Tight Linearization and an Algorithm for Zero-One Quadratic Programming Problems," Management Science, INFORMS, vol. 32(10), pages 1274-1290, October.
    5. Peter Hahn & Thomas Grant, 1998. "Lower Bounds for the Quadratic Assignment Problem Based upon a Dual Formulation," Operations Research, INFORMS, vol. 46(6), pages 912-922, December.
    6. Fred Glover, 1975. "Improved Linear Integer Programming Formulations of Nonlinear Integer Problems," Management Science, INFORMS, vol. 22(4), pages 455-460, December.
    7. H. D. Sherali & R. S. Krishnamurthy & F. A. Al-Khayyal, 1998. "Enumeration Approach for Linear Complementarity Problems Based on a Reformulation-Linearization Technique," Journal of Optimization Theory and Applications, Springer, vol. 99(2), pages 481-507, November.
    8. Fred Glover & Eugene Woolsey, 1974. "Technical Note—Converting the 0-1 Polynomial Programming Problem to a 0-1 Linear Program," Operations Research, INFORMS, vol. 22(1), pages 180-182, February.
    9. Warren P. Adams & Hanif D. Sherali, 1990. "Linearization Strategies for a Class of Zero-One Mixed Integer Programming Problems," Operations Research, INFORMS, vol. 38(2), pages 217-226, April.
    10. Hanif D. Sherali & Warren P. Adams & Patrick J. Driscoll, 1998. "Exploiting Special Structures in Constructing a Hierarchy of Relaxations for 0-1 Mixed Integer Problems," Operations Research, INFORMS, vol. 46(3), pages 396-405, June.
    11. Jean B. Lasserre, 2002. "Semidefinite Programming vs. LP Relaxations for Polynomial Programming," Mathematics of Operations Research, INFORMS, vol. 27(2), pages 347-360, May.
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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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