Integrality of Subgradients and Biconjugates of Integrally Convex Functions

TL;DR

This paper proves that integrally convex functions have integral subgradients and their biconjugates coincide with themselves, unifying various classes of discrete convex functions and extending discrete DC function theory.

Contribution

It establishes integral subdifferentiability and biconjugacy for integrally convex functions, unifying multiple classes of discrete convex functions under a common framework.

Findings

Integral subgradients exist for integer-valued integrally convex functions.

The integral biconjugate of such functions equals the original function.

Unified proof for integral biconjugacy across various discrete convex function classes.

Abstract

Integrally convex functions constitute a fundamental function class in discrete convex analysis. This paper shows that an integer-valued integrally convex function admits an integral subgradient and that the integral biconjugate of an integer-valued integrally convex function coincides with itself. The proof is based on the Fourier-Motzkin elimination. The latter result provides a unified proof of integral biconjugacy for various classes of integer-valued discrete convex functions, including L-convex, M-convex, L$_{2}$-convex, M$_{2}$-convex, BS-convex, and UJ-convex functions as well as multimodular functions. Our results of integral subdifferentiability and integral biconjugacy make it possible to extend the theory of discrete DC (difference of convex) functions developed for L- and M-convex functions to that for integrally convex functions, including an analogue of the Toland--Singer duality for integrally convex functions.