Convex Analysis and Optimization with Submodular Functions: a Tutorial

TL;DR

This tutorial provides a comprehensive, self-contained overview of submodular functions, highlighting their importance in various fields and drawing parallels with convex functions, aimed at readers with a background in convex analysis.

Contribution

It offers a detailed, foundational presentation of submodular functions from first principles, serving as an accessible resource for understanding their theory and applications.

Findings

Submodular functions are analogous to convex functions on sets.

The tutorial covers core properties and results of submodular functions.

It bridges concepts from convex analysis to set functions.

Abstract

Set-functions appear in many areas of computer science and applied mathematics, such as machine learning, computer vision, operations research or electrical networks. Among these set-functions, submodular functions play an important role, similar to convex functions on vector spaces. In this tutorial, the theory of submodular functions is presented, in a self-contained way, with all results shown from first principles. A good knowledge of convex analysis is assumed.