A Survey of Fundamental Operations on Discrete Convex Functions of Various Kinds

TL;DR

This survey reviews fundamental operations on various types of discrete convex functions, highlighting their roles across multiple fields like operations research, game theory, and economics.

Contribution

It provides a comprehensive overview of the fundamental operations applicable to different classes of discrete convex functions in discrete convex analysis.

Findings

Summarizes key operations for integrally convex, L-convex, M-convex, and multimodular functions.

Highlights the applications of these functions in diverse fields.

Identifies open problems and future research directions.

Abstract

Discrete convex functions are used in many areas, including operations research, discrete-event systems, game theory, and economics. The objective of this paper is to offer a survey on fundamental operations for various kinds of discrete convex functions in discrete convex analysis such as integrally convex functions, L-convex functions, M-convex functions, and multimodular functions.