Nonlinear Discrete Optimization

Overview

In these lectures we develop an emerging algorithmic theory of nonlinear discrete optimization based on exciting recent advances over the last few years, extending the well-established theory of linear discrete optimization.

We will cover some of the highlights of this theory including efficient algorithms for maximization of convex functions over sets of integer points presented by inequalities or oracles; maximization and minimization of convex functions over integer n-fold systems and integer stochastic systems; efficient randomized and approximative optimization of arbitrary nonlinear functions over combinatorial systems; and the universality theorem for rational polytopes and their integer points.

We use a variety of geometric and algebraic methods, some classical and some very recent, that will be developed as part of the lectures, such as the theory of Graver bases and their stabilization over n-fold systems and stochastic systems, zonotopes, unimodular matrices, Frobenius numbers, and polynomial identity testing and interpolation.

We will also discuss some of the many applications of this theory in statistics, operations research, and commutative algebra, including privacy in statistical databases and experimental design; nonlinear transportation and multicommodity flows; error-correcting codes and construction of universal Grobner bases on the Hilbert scheme.

The lectures are accessible to anyone with standard undergraduate knowledge, in particular linear algebra (though some exposure to basic complexity and linear programming may help intuition); the only main real requirement is mathematical maturity.

The Nachdiplom Lecture Notes will be posted below as the lectures evolve. They will extend my recent monograph Convex Discrete Optimization, and will eventually be incorporated into our research book on the subject (coauthored with Jon Lee and Robert Weismantel) under preparation.

Lecture notes

Nachdiplom Lecture Notes

The course is based on our research monograph under preparation, which extends my recent monograph Convex Discrete Optimization. You can download it from arXiv.

Exercises

All exercises are due two weeks after the publishing date. Please hand the exercises in on a recitation, or in the "Discrete Optimization" tray at HG G 21.

Requirements

For Bachelor and Master students: Correctly solving 50% of the homework exercises to qualify for the exam.

For PhD Students: Correctly solving 50% of the exercises. Pass/Fail evaluation.

Exam: 20 Minutes Oral exam.

Dates

• The course will start on March 4, 2009.
• The final exam will take place during the first week of June 2009.