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An Implementation of the Generalized Basis Reduction Algorithm for Integer Programming

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Author Info
William Cook
Thomas Rutherford
Herbert E. Scarf () (Cowles Foundation, Yale University)
David F. Shallcross

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Abstract

In recent years many advances have been made in solution techniques for specially structured 0-1 integer programming problems. In contrast, very little progress has been made on solving general (mixed integer) problems. This, of course, is not true when viewed from the theoretical side: Lenstra (1981) made a major breakthrough, obtaining a polynomial-time algorithm when the number of integer variables is fixed. We discuss a practical implementation of a Lenstra-like algorithm, based on the generalized basis reduction method of Lovasz and Scarf (1988).This method allows us to avoid the ellipsoidal approximations required in Lenstra's algorithm. We report on the solution of a number of small (but difficult) examples, up to 100 integer variables. Our computer code uses the linear programming optimizer CPlex as a subroutine to solve the linear programming problems that arise.

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Publisher Info
Paper provided by Cowles Foundation, Yale University in its series Cowles Foundation Discussion Papers with number 990.

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Length: 13 pages
Date of creation: Aug 1991
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Publication status: Published in ORSA Journal of Computing (spring 1993), 5(2): 206-221
Handle: RePEc:cwl:cwldpp:990

Note: CFP 906.
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Postal: Yale University, Box 208281, New Haven, CT 06520-8281 USA
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Related research
Keywords: Linear programming; mixed integer problems;

Find related papers by JEL classification:
C61 - Mathematical and Quantitative Methods - - Mathematical Methods and Programming - - - Optimization Techniques; Programming Models; Dynamic Analysis
C63 - Mathematical and Quantitative Methods - - Mathematical Methods and Programming - - - Computational Techniques

References listed on IDEAS
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  1. Herbert E. Scarf & Laszlo Lovasz, 1990. "The Generalized Basis Reduction Algorithm," Cowles Foundation Discussion Papers 946, Cowles Foundation, Yale University. [Downloadable!]
  2. Herbert E. Scarf & R. Kannan & Laszlo Lovasz, 1988. "The Shapes of Polyhedra," Cowles Foundation Discussion Papers 883, Cowles Foundation, Yale University. [Downloadable!]
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