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

MATHEMATICAL programming 2011年第1期130卷 153-176页

作者： Zanette, Arrigo Fischetti, Matteo Balas, Egon Univ Padua DEI Padua Italy Carnegie Mellon Univ Pittsburgh PA 15213 USA

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.

关键词： Cutting plane methods Gomory cuts degeneracy in linear programming Lexicographic dual simplex Computational analysis

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A hard integer program made easy by lexicography

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MATHEMATICAL programming 2012年第1-2期135卷 509-514页

作者： Balas, Egon Fischetti, Matteo Zanette, Arrigo Univ Padua DEI Padua Italy Carnegie Mellon Univ Pittsburgh PA 15213 USA

A small but notoriously hard integer program formulated by Donald Knuth fifty years ago is solved by three versions of a lexicographic algorithm using Gomory cuts. The lexicographic cutting plane algorithms are faster... 详细信息

关键词： Cutting plane methods Gomory cuts degeneracy in linear programming Lexicographic dual simplex Computational analysis

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On the enumerative nature of Gomory's dual cutting plane method

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MATHEMATICAL programming 2010年第2期125卷 325-351页

作者： Balas, Egon Fischetti, Matteo Zanette, Arrigo Univ Padua DEI Padua Italy Carnegie Mellon Univ Pittsburgh PA 15213 USA

For 30 years after their invention half a century ago, cutting planes for integer programs have been an object of theoretical investigations that had no apparent practical use. When they finally proved their practical usefulness in the late eighties, that happened in the framework of branch and bound procedures, as an auxiliary tool meant to reduce the number of enumerated nodes. To this day, pure cutting plane methods alone have poor convergence properties and are typically not used in practice. Our reason for studying them is our belief that these negative properties can be understood and thus remedied only based on a thorough investigation of such procedures in their pure form. In this paper, the second in a sequence, we address some important issues arising when designing a computationally sound pure cutting plane method. We analyze the dual cutting plane procedure proposed by Gomory in 1958, which is the first (and most famous) convergent cutting plane method for integer linear programming. We focus on the enumerative nature of this method as evidenced by the relative computational success of its lexicographic version (as documented in our previous paper on the subject), and we propose new versions of Gomory's cutting plane procedure with an improved performance. In particular, the new versions are based on enumerative schemes that treat the objective function implicitly, and redefine the lexicographic order on the fly to mimic a sound branching strategy. Preliminary computational results are reported.

关键词： Cutting plane methods Gomory cuts degeneracy in linear programming Lexicographic dual simplex Computational analysis

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On the enumerative nature of Gomory's dual cutting plane method

On the enumerative nature of Gomory's dual cutting plane met...

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20th International Symposium of Mathematical programming (ISMP)

作者： Balas, Egon Fischetti, Matteo Zanette, Arrigo Univ Padua DEI Padua Italy Carnegie Mellon Univ Pittsburgh PA 15213 USA

关键词： Cutting plane methods Gomory cuts degeneracy in linear programming Lexicographic dual simplex Computational analysis

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Can pure cutting plane algorithms work?

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13th International Conference on Integer programming and Combinatorial Optimization

作者： Zanette, Arrigo Fischetti, Matteo Balas, Egon Univ Padua DEI I-35100 Padua Italy Carnegie Mellon Univ Pittsburgh PA 15213 USA

ISBN: (纸本)9783540688860

We discuss an implementation of the lexicographic version of Gomory's fractional cutting plane method 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 surprizing phenomenon.

关键词： cutting plane methods Gomory cuts degeneracy in linear programming lexicographic dual simplex computational analysis

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