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Mathematical Programming

, Volume 94, Issue 2–3, pp 221–245 | Cite as

A precise correspondence between lift-and-project cuts, simple disjunctive cuts, and mixed integer gomory cuts for 0-1 programming

  • Egon Balas
  • Michael Perregaard

Abstract.

 We establish a precise correspondence between lift-and-project cuts for mixed 0-1 programs, simple disjunctive cuts (intersection cuts) and mixed-integer Gomory cuts. The correspondence maps members of one family onto members of the others. It also maps bases of the higher-dimensional cut generating linear program (CGLP) into bases of the linear programming relaxation. It provides new bounds on the number of facets of the elementary closure, and on the rank, of the standard linear programming relaxation of the mixed 0-1 polyhedron with respect to the above families of cutting planes.

Based on the above correspondence, we develop an algorithm that solves (CGLP) without explicitly constructing it, by mimicking the pivoting steps of the higher dimensional (CGLP) simplex tableau by certain pivoting steps in the lower dimensional (LP) simplex tableau. In particular, we show how to calculate the reduced costs of the big tableau from the entries of the small tableau and based on this, how to identify a pivot in the small tableau that corresponds to one or several improving pivots in the big tableau. The overall effect is a much improved lift-and-project cut generating procedure, which can also be interpreted as an algorithm for a systematic improvement of mixed integer Gomory cuts from the small tableau.

Keywords

Mixed Integer Systematic Improvement Linear Programming Relaxation Elementary Closure Precise Correspondence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Egon Balas
    • 1
  • Michael Perregaard
    • 2
  1. 1.Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA; e-mail: eb17@andrew.cmu.eduUS
  2. 2.Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA; e-mail: michael4@andrew.cmu.eduUS

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