Shmuel Onn
Nonlinear Discrete Optimization
Zurich Lectures in Advanced Mathematics,
European Mathematical Society, 2010, 147 pages

This monograph develops an algorithmic theory of nonlinear discrete optimization. It introduces a simple and useful setup which enables the polynomial time solution of broad fundamental classes of nonlinear combinatorial optimization and integer programming problems in variable dimension. An important part of this theory is enhanced by recent developments in the algebra of Graver bases. The power of the theory is demonstrated by deriving the first polynomial time algorithms in a variety of application areas within operations research and statistics, including vector partitioning, matroid optimization, experimental design, multicommodity flows, multi-index transportation and privacy in statistical databases.

Preliminary partial notes, Nachdiplom Lectures, FIM ETH Zürich, Spring 2009:


1. Introduction

2. From Linear to Convex Maximization

3. Graver Bases and Nonlinear Integer Programming

4. N-Fold Integer Programming

5. Multiway Tables and Universality

6. Nonlinear Combinatorial Optimization