Binary extended formulations and sequential convexification

TL;DR

This paper introduces a unified framework for binary extended formulations in mixed-integer programming, characterizes their vertices, and analyzes their behavior under sequential convexification to improve convergence towards integrality.

Contribution

It defines a notion of natural binarizations, characterizes vertices of these formulations, and studies their sequential convexification behavior, providing a new perspective on convergence properties.

Findings

Characterization of vertices of natural binary extended formulations.

Introduction of a rank parameter to measure progress in convexification.

Exact expression of the rank for classical binarizations.

Abstract

A binarization of a bounded variable $x$ is a linear formulation with variables $x$ and additional binary variables $y_1,\dots, y_k$, so that integrality of $x$ is implied by the integrality of $y_1,\dots, y_k$. A binary extended formulation of a polyhedron $P$ is obtained by adding to the original description of $P$ binarizations of some of its variables. In the context of mixed-integer programming, imposing integrality on 0/1 variables rather than on general integer variables has interesting convergence properties and has been studied both from the theoretical and from the practical point of view. We propose a notion of \emph{natural} binarizations and binary extended formulations, encompassing all the ones studied in the literature. We give a simple characterization of the vertices of such formulations, which allows us to study their behavior with respect to sequential convexification. %0/1 disjunctions. In particular, given a binary extended formulation and % a binarization $B$ of one of its variables $x$, we study a parameter that measures the progress made towards ensuring the integrality of $x$ via application of sequential convexification. We formulate this parameter, which we call rank, as the solution of a set covering problem and express it exactly for the classical binarizations from the literature.