Scaling, Proximity, and Optimization of Integrally Convex Functions

TL;DR

This paper investigates the properties of integrally convex functions, revealing that scaling only holds in low dimensions, but proximity theorems can be extended, enabling efficient minimization algorithms for fixed-variable cases.

Contribution

It demonstrates the limitations of scaling properties in higher dimensions and extends proximity results, facilitating optimization of integrally convex functions with fixed variables.

Findings

Scaling property holds only for n ≤ 2 variables.

Proximity theorem exists for any n but with superexponential bounds.

Efficient minimization algorithms extend from 1D to fixed-variable cases.

Abstract

In discrete convex analysis, the scaling and proximity properties for the class of L$^\natural$-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of $n$ variables, we show here that the scaling property only holds when $n \leq 2$, while a proximity theorem can be established for any $n$, but only with a superexponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discrete convex function of one variable to the case of integrally convex functions of any fixed number of variables.