Foundations of Discrete Optimization: In Transition from Linear to Non-linear Models and Methods

Abstract

Optimization is a vibrant growing area of Applied Mathematics. Its many successful applications depend on efficient algorithms and this has pushed the development of theory and software. In recent years there has been a resurgence of interest to use “non-standard” techniques to estimate the complexity of computation and to guide algorithm design. New interactions with fields like algebraic geometry, representation theory, number theory, combinatorial topology, algebraic combinatorics, and convex analysis have contributed non-trivially to the foundations of computational optimization. In this expository survey we give three example areas of optimization where “algebraic-geometric thinking” has been successful. One key point is that these new tools are suitable for studying models that use non-linear constraints together with combinatorial conditions.