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Logical Reduction Methods in Zero-One Programming—Minimal Preferred Variables

Author

Listed:
  • Monique Guignard

    (University of Pennsylvania, Philadelphia, Pennsylvania)

  • Kurt Spielberg

    (IBM Corporation, White Plains, New York)

Abstract

In the first part of this paper, the concept of logical reduction is presented Minimal preferred variable inequalities are introduced, and algorithms are given for their calculation. A simple illustrative example is carried along from the start, further examples are provided later. The second part of the paper describes certain properties of the generated logical inequalities. It then explains some of the decreases of computational effort which may be achieved by the use of minimal preferred inequalities and outlines a number of concrete applications with some numerical results. Finally, a number of more recent concepts and results are discussed, among them the notion of “probing” and a related zero-one enumeration code for large scale problems under the extended control language of MPSX/370.

Suggested Citation

  • Monique Guignard & Kurt Spielberg, 1981. "Logical Reduction Methods in Zero-One Programming—Minimal Preferred Variables," Operations Research, INFORMS, vol. 29(1), pages 49-74, February.
    • Handle: RePEc:inm:oropre:v:29:y:1981:i:1:p:49-74
    DOI: 10.1287/opre.29.1.49
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    Cited by:

    1. Patrick Gemander & Wei-Kun Chen & Dieter Weninger & Leona Gottwald & Ambros Gleixner & Alexander Martin, 2020. "Two-row and two-column mixed-integer presolve using hashing-based pairing methods," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 8(3), pages 205-240, October.
    2. Escudero Bueno, Laureano F. & Garín Martín, María Araceli & Merino Maestre, María & Pérez Sainz de Rozas, Gloria, 2011. "A parallelizable algorithmic framework for solving large scale multi-stage stochastic mixed 0-1 problems under uncertainty," BILTOKI 1134-8984, Universidad del País Vasco - Departamento de Economía Aplicada III (Econometría y Estadística).
    3. L. Escudero & A. Garín & G. Pérez, 1996. "Some properties of cliques in 0–1 mixed integer programs," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 4(2), pages 215-223, December.
    4. 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.
    5. Tobias Achterberg & Robert E. Bixby & Zonghao Gu & Edward Rothberg & Dieter Weninger, 2020. "Presolve Reductions in Mixed Integer Programming," INFORMS Journal on Computing, INFORMS, vol. 32(2), pages 473-506, April.
    6. Robert Ashford, 2007. "Mixed integer programming: A historical perspective with Xpress-MP," Annals of Operations Research, Springer, vol. 149(1), pages 5-17, February.
    7. Ambros Gleixner & Leona Gottwald & Alexander Hoen, 2023. "P a PILO: A Parallel Presolving Library for Integer and Linear Optimization with Multiprecision Support," INFORMS Journal on Computing, INFORMS, vol. 35(6), pages 1329-1341, November.

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