The structure of the infinite models in integer programming

TL;DR

This paper explores the relationship between two descriptions of infinite models in integer programming, revealing that nonnegative, continuous valid functions are sufficient to describe corner polyhedra, impacting their theoretical understanding.

Contribution

It establishes the equivalence between convex hull and halfspace intersection descriptions for infinite models, especially for corner polyhedra, using nonnegative, continuous valid functions.

Findings

Nonnegative, continuous valid functions suffice for corner polyhedra.

The relationship between convex hull and halfspace descriptions is clarified.

Implications for the structure of corner polyhedra are discussed.

Abstract

The infinite models in integer programming can be described as the convex hull of some points or as the intersection of halfspaces derived from valid functions. In this paper we study the relationships between these two descriptions. Our results have implications for corner polyhedra. One consequence is that nonnegative, continuous valid functions suffice to describe corner polyhedra (with or without rational data).