Abstract

It is common knowledge that the understanding of the combinatorial geometry of convex bodies has helped speed up algorithms in discrete optimization. For example, cutting planes and facet-description of polyhedra have been crucial in the success of branch-and-bound algorithms for mixed integer linear programming. Another example, is how the ellipsoid method can be used to prove polynomiality results in combinatorial optimization. For the future, the importance of algebraic-combinatorial geometry in optimization appears even greater.

In the past 5 years advances in algebraic-geometric algorithms have been used to prove unexpected new results on the computation of non-linear integer programs. These lectures will introduce the audience to new techniques. I will describe several algorithms and explain why we can now prove theorems that were beyond our reach before, mostly about integer optimization with non-linear objectives. I will also describe attempts to turn these two algorithms into practical computation, not just in theoretical results.

This a nice story collecting results by various authors and now contained in our monograph recently published by SIAM -MOS.