Summer school EUR MINT 2022 Moments & Positive Polynomials & their Applications

27 juin 2022, 09:00 → 30 juin 2022, 16:00 Europe/Paris

Amphithéâtre Schwartz (Institut de Mathématiques)

This short course proposal targets Master or Doctoral students as well as
researchers and engineers who could be interested.

It provides them with a first introduction to certain tools of algebraic geometry and in particular the theory of positive polynomials and its dual, the K-moment problem.

This has many important applications not only in polynomial optimization but
also in many other areas (probability, statistics, dynamical systems, ODEs
and PDEs). These tools have already had an impact in several important
applications (e.g. in optimization, signal processing (Super-Resolution),
Optimal Design in statistics, control, computational geometry) and some
Machine Learning applications (computer vision, tensor decomposition &
completion, dictionary learning, mixture of Gaussians). In addition, noncommutative analogues of such tools are becoming of crucial importance in quantum information.

The course will be a 15  hours module, spread out over 4 days.

Speakers:

• Jean-Bernard Lasserre (LAAS-CNRS)
• Milan Korda (LAAS-CNRS)
• Victor Magron (LAAS-CNRS)

lundi 27 juin

1 Basics of Conic Optimization (J.-B. Lasserre) SOS and semidefinite programming (J.-B. Lasserre)

10:30 Coffee break

2 Positivity certificates and K-moment problem (J.-B. Lasserre)

12:00 Lunch break

3 LP- and SOS-based hierarchy for optimization (J.-B. Lasserre)

15:00 Coffee break

4 Comparing LP and SOS-based hierarchies – Finite convergence & Global Optimality conditions (J.-B. Lasserre

mardi 28 juin

5 Some applications of the Moment Problem (J.-B. Lasserre)

10:30 Coffee break

6 Exploiting correlative sparsity (Victor Magron)

12:00 Lunch break

7 Exploiting term sparsity (Victor Magron)

15:00 Coffee break

8 Certified optimization: from approximate to exact bounds (Victor Magron)

mercredi 29 juin

9 Noncommutative optimization & quantum information (Victor Magron)

10:30 Coffee break

10 Exercise session (Victor Magron)

12:00 Lunch break

11 Occupation measures for dynamical systems & control (Milan Korda)

15:00 Coffee break

12 Trajectory recovery (atomic approximation, Christoffel-Darboux Kernel

jeudi 30 juin

13 The region of attraction, invariant sets and extreme values (Milan Korda)

10:30 Coffee break

14 Extensions: Partial differential equations, complexity reduction (sparsity, state-space partition) (Milan Korda)

12:00 Lunch break

15 Exercises (Milan Korda)