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

• Published in: Mathematical programming Vol. 125; no. 2; pp. 325 - 351
• Main Authors: Balas, Egon, Fischetti, Matteo, Zanette, Arrigo
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.10.2010; Springer; Springer Nature B.V
• Subjects: Algorithms; Applied sciences; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Computation; Computational mathematics; Convergence; Cutting; Exact sciences and technology; Full Length Paper; Integer programming; Integers; 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; Sound; Strategy; 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; Belief; Branching; Pivoting method; Computational analysis 90C 10 Integer programming; Multicriteria analysis; Branch and bound method; Degenerate system; Linear programming; Cutting plane method; Cut generation; Integer programming; Extremum; Lexicography; Lexicographic order; Objective function; Mathematical programming
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-010-0392-4

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.