Tóm tắt bài giảng

1. Krzysztof Kurdyka                                                                                                 

Lecture 1

Title: Composed univariate sum of squares approximation of polynomials (Joint work with S. Spodzieja). 

Abstract: We show that if a polynomial is nonnegative on a closed basic semialgebraic set where, then f can be approximated uniformly on compact sets by polynomials of the form, where and are sums of squares of polynomials. In particular, if X icompact, and is positive on X, then for some sums of squares and, where.  

Lecture 2

Title: Polynomial and exponential convexifying of positive polynomials (Joint work with K. Rudnicka and S. Spodzieja). 

Abstract: Let be a convex closed and semialgebraic set and let f be a polynomial positive on X. We prove that there exists an exponent, such that for any the function is strongly convex on X. When X is unbounded we have to assume also that the leading form of f is positive in. We obtain strong convexity of on possibly unbounded X, provided N is sufficiently large, assuming only that f is positive on X. We apply these results for searching critical points of polynomials on convex closed semialgebraic sets.  


2. Jean Bernard Lasserre 

Title: The Moment-SOS hierarchy 


  1. One about the Moment-SOS hierarchy where I introduce the Moemnt-SOS hierarchy, positivity certificates, semidefinite relaxations, sparsity, etc.
  2. One where I describe a few applications outside optimization (Optimal control, non-linear hyperbolic PDEs', super-resolution, polynomial interpolation, etc


3. Konrad Schmudgen

Lectures on the Moment Problem Konrad Schm¨udgen (University of Leipzig)

Lecture 1:Integral representation of linear functionals An integral representation theorem of positive functionals on Choquet’s adapted spaces is obtained. As applications, Haviland’s theorem is derived and existence results for moment problems on intervals are developed. Moment problems of _-semigroups are briefly discussed.

Lecture 2. Multidimensional moment problem on compact semi-algebraic sets The interplay between the moment problem and Positivstellens¨atze of real algebraic geometry on compact semi-algebraic sets is developed. Existence criteria for special semi-algebraic sets are given.

Lecture3: Polynomial optimization and semidefinite programming Semidefinite programming is introduced and the Lasserre relaxations are defined. From the Archimedean Positivstellensatz convergence results are obtained.

Lecture 4: Truncated multidimensional moment problem: existence via positivity The theorem of Richter-Tchakaloff is proved. Existence criteria by positivity conditions are formulated. Stochel’s theorem is mentioned.

Lecture 5: Truncated multidimensional moment problem: existence via flat extension Hankel matrices and their basic properties are treated. The flat extension theorem of Curto and Fialkow is stated and explained.

Lecture 6: Truncated multidimensional moment problem: the moment cone The core variety is defined and basic results about the core variety are obtained. Results on the structure of the moment cone are discussed.

Download lecture

4. Phạm Tiến Sơn

Title: Optimization of polynomials on basic closed semi-algebraic sets 


In this lecture, we address the problem of minimizing a polynomial function over a basic closed semi-algebraic set. The cases where the constraint set is bounded and unbounded are considered. In each case we construct a sequence of semidefinite programs whose optimal values converge monotonically, increasing to the optimal value of the original problem. Finally, some generic properties of polynomial optimization problems are also discussed.


5. Grzegorz Oleksik

Lecture 1:

Title: Conjecture on the Łojasiewicz exponent.


In this talk, we propose a conjecture that the Łojasiewicz exponent of a nondegenerate  (in the Kouchnirenko sense)  isolated singularity (complex case) could be read off from its Newton diagram. We also discuss the history and progress of this problem.

Lecture 2:

Title: The Łojasiewicz Exponent at infinity of non-negative and non-degenerate polynomials


Let f be a real polynomial, non-negative at infinity with non-compact zeroset. Suppose that f is non-degenerate in theKushnirenko sense at infinity. In this talk we give a formula for the Łojasiewicz exponent at infinity of f and a formula for the exponent of growth of f in terms of its Newton polyhedron.