Another pedagogy for mixed-integer Gomory

TL;DR

This paper introduces a novel approach to deriving Gomory mixed-integer cuts using a dual form perspective and primal simplex columns, enabling a finitely converging algorithm for mixed-integer optimization.

Contribution

It develops a new method for generating GMI cuts from a dual form problem, differing from traditional approaches, and implements a primal-simplex column-generation algorithm.

Findings

New GMI cut derivation from dual form problems.

Finitely converging primal-simplex column-generation algorithm.

Applicable to non-standard form mixed-integer problems.

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.