## Japan Journal of Industrial and Applied Mathematics

### A Note on Discrete Convexity and Local Optimality

Takashi Ui

#### Abstract

One of the most important properties of a convex function is that a local optimum is also a global optimum. This paper explores the discrete analogue of this property. We consider arbitrary locality in a discrete space and the corresponding local optimum of a function over the discrete space. We introduce the corresponding notion of discrete convexity and show that the local optimum of a function satisfying the discrete convexity is also a global optimum. The special cases include discretely-convex, integrally-convex, M-convex, $\text{M}^\natural$-convex, L-convex, and $\text{L}^\natural$-convex functions.

#### Article information

Source
Japan J. Indust. Appl. Math., Volume 23, Number 1 (2006), 21-29.

Dates
First available in Project Euclid: 19 June 2006