Recent Progress on Integrally Convex Functions

TL;DR

This paper provides a comprehensive survey of recent advances in integrally convex functions, highlighting new technical results and covering topics like characterizations, operations, optimality criteria, and duality in discrete convex analysis.

Contribution

It offers a detailed overview of recent developments in integrally convex functions, including new theoretical insights and technical results that extend the existing framework.

Findings

Characterizations of integral convex sets and functions

Operations on integral convex sets and functions

Optimality criteria for minimization with proximity-scaling algorithm

Abstract

Integrally convex functions constitute a fundamental function class in discrete convex analysis, including M-convex functions, L-convex functions, and many others. This paper aims at a rather comprehensive survey of recent results on integrally convex functions with some new technical results. Topics covered in this paper include characterizations of integral convex sets and functions, operations on integral convex sets and functions, optimality criteria for minimization with a proximity-scaling algorithm, integral biconjugacy, and the discrete Fenchel duality. While the theory of M-convex and L-convex functions has been built upon fundamental results on matroids and submodular functions, developing the theory of integrally convex functions requires more general and basic tools such as the Fourier-Motzkin elimination.