In the past fifteen years, methods from algebra and algebraic geometry have been used in optimization to design algorithms and understand the structure of optimization problems. These techniques include basis reduction, the theory of Groebner bases, rational generating functions, nonnegativity of real polynomials and further methods from real algebraic geometry. This tutorial will be roughly divided into two parts---the first focussing on algebraic methods in discrete optimization and the second on methods from algebra and real algebraic geometry in semi-definite programming and polynomial optimization.

The tutorial will provide an introduction to these methods and will be aimed at non-specialists. The activities will consist of introductory lectures followed by hands-on computational sessions using state-of-the-art software packages in these fields.