Polynomial Optimization Methods

This research is an exposition of ideas and methods that help understanding the problem of minimizing a polynomial over a basic closed semi-algebraic set. After the introduction of some theory on mathematical tools such as sums of squares, nonnegative polynomials and moment matrices, several Positivstellensatze are considered. Positivstellensatze provide sums of squares representations of polynomials, positive on basic closed semi-algebraic sets. Subsequently, semi-de_nite programming methods, in particular based on Putinar's Postivstellensatz, are considered. In order to use semi-de_nite programming, certain degree bounds are set. These bounds give rise to a hierarchy of approximations of the minimum of a polynomial, which will also be discussed. Finally, some new results are given that are obtained by looking at sums of squares representations of a positive polynomial when minimizing over the unit hypercube.