Japan Journal of Industrial and Applied Mathematics

A Note on Discrete Convexity and Local Optimality

Takashi Ui

Full-text: Open access


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

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

First available in Project Euclid: 19 June 2006

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

discrete optimization convex function quasiconvex function Nash equilibrium potential game


Ui, Takashi. A Note on Discrete Convexity and Local Optimality. Japan J. Indust. Appl. Math. 23 (2006), no. 1, 21--29. https://projecteuclid.org/euclid.jjiam/1150725469

Export citation