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Another pedagogy for mixed-integer Gomory

Jon Lee(jonxlee***at***umich.edu)
Angelika Wiegele(angelika.wiegele***at***aau.at)

Abstract: We present a version of GMI (Gomory mixed-integer) cuts in a way so that they are derived with respect to a ``dual form'' mixed-integer optimization problem and applied on the standard-form primal side as columns, using the primal simplex algorithm. This follows the general scheme of He and Lee, who did the case of Gomory pure-integer cuts. Our input mixed-integer problem is not in standard form, and so our cuts are derived rather differently from how they are normally derived. A convenient way to develop GMI cuts is from MIR (mixed-integer rounding) cuts, which are developed from 2-dimensional BMI (basic mixed integer) cuts, which involve a nonnegative continuous variable and an integer variable. The nonnegativity of the continuous variable is not the right tool for us, as our starting point (the ``dual form'' mixed-integer optimization problem) has no nonnegativity. So we work out a different 2-dimensional starting point, a pair of somewhat arbitrary inequalities in one continuous and one integer variable. In the end, we follow the approach of He and Lee, getting now a finitely converging primal-simplex column-generation algorithm for mixed-integer optimization problems.

Keywords: mixed-integer programming, Gomory mixed-integer cut, mixed-integer rounding cut, basic mixed-integer cut, column generation, primal simplex algorithm

Category 1: Integer Programming ((Mixed) Integer Linear Programming )

Category 2: Integer Programming (Cutting Plane Approaches )

Citation: arXiv 1510.06522 http://arxiv.org/abs/1510.06522

Download: [PDF]

Entry Submitted: 10/25/2015
Entry Accepted: 10/25/2015
Entry Last Modified: 10/25/2015

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