Markus Schweighofer
University of Konstanz
Part 1: Polya's Theorem and Representation of Positive Polynomials
1. Polya's Theorem
2. Proving Putinar's Positivstellensatz using Polya's Theorem
3. A less general approach which allows to keep track of complexity
4. Discussion: Relationship between the Positivstellensätze of
Krivine 1964 (=Stengle 1974), Schmüdgen 1991 and Putinar 1993
Part 2: Optimization of polynomials on compact basic closed semialgebraic sets
1. Convexification by brute force: Probability measures instead
of points or dual standpoint (positivity)
2. Putinar's solution to the moment problem and
Putinar's Positivstellensatz
3. Motivation and formulation of Lasserre's relaxations
in an abstract way (i.e., without speaking about matrices)
(equality constraint treated as two inequalities).
4. Convergence against the minimum value.
5. Speed of convergence: Complexity of Schmüdgen's
Positivstellensatz
6. Convergence of the first order moments against unique
minimizers.
7. A geometric interpretation of duality, very rough sketch of
algebraic proof of duality.