Lexicography and degeneracy: can a pure cutting plane algorithm work?

• Published in: Mathematical programming Vol. 130; no. 1; pp. 153 - 176
• Main Authors: Zanette, Arrigo, Fischetti, Matteo, Balas, Egon
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-11-2011; Springer; Springer Nature B.V
• Subjects: Algorithms; Applied sciences; Battery; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Cutting; Exact sciences and technology; Exact solutions; Full Length Paper; Integer programming; Linear programming; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematical programming; Mathematics; Mathematics and Statistics; Mathematics of Computing; Methods; Numerical Analysis; Operational research and scientific management; Operational research. Management science; Planes; Radicals; Studies; Theoretical; Cutting plane methods; 90C10 Integer programming; Computational analysis; Degeneracy in linear programming; 90C05 Linear programming; Lexicographic dual simplex; Gomory cuts; 90C49 Extreme-point and pivoting methods; Degeneracy in linear programing; Pivoting method; Degenerate system; Linear programming; Secondary cell; Modeling; Cutting plane method; Cut generation; Computational analysis 90C10 Integer programming; Standards; Exact solution; Integer programming; Extremum; Lexicography; Heuristic method; Mathematical programming
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-009-0335-0

We discuss an implementation of the lexicographic version of Gomory’s fractional cutting plane method for ILP problems and of two heuristics mimicking the latter. In computational testing on a battery of MIPLIB problems we compare the performance of these variants with that of the standard Gomory algorithm, both in the single-cut and in the multi-cut (rounds of cuts) version, and show that they provide a radical improvement over the standard procedure. In particular, we report the exact solution of ILP instances from MIPLIB such as stein15, stein27, and bm23, for which the standard Gomory cutting plane algorithm is not able to close more than a tiny fraction of the integrality gap. We also offer an explanation for this surprising phenomenon.