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

Europe/Paris
Amphithéâtre Schwartz (Institut de Mathématiques)

Amphithéâtre Schwartz

Institut de Mathématiques

Université Toulouse 3 Paul Sabatier 118 Route de Narbonne Institut de Mathématiques- Bâtiment 1R3 Toulouse
Description

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)
    • 1
      Basics of Conic Optimization (J.-B. Lasserre) SOS and semidefinite programming (J.-B. Lasserre)
    • 10:30 AM
      Coffee break
    • 2
      Positivity certificates and K-moment problem (J.-B. Lasserre)
    • 12:00 PM
      Lunch break
    • 3
      LP- and SOS-based hierarchy for optimization (J.-B. Lasserre)
    • 3:00 PM
      Coffee break
    • 4
      Comparing LP and SOS-based hierarchies – Finite convergence & Global Optimality conditions (J.-B. Lasserre
    • 5
      Some applications of the Moment Problem (J.-B. Lasserre)
    • 10:30 AM
      Coffee break
    • 6
      Exploiting correlative sparsity (Victor Magron)
    • 12:00 PM
      Lunch break
    • 7
      Exploiting term sparsity (Victor Magron)
    • 3:00 PM
      Coffee break
    • 8
      Certified optimization: from approximate to exact bounds (Victor Magron)
    • 9
      Noncommutative optimization & quantum information (Victor Magron)
    • 10:30 AM
      Coffee break
    • 10
      Exercise session (Victor Magron)
    • 12:00 PM
      Lunch break
    • 11
      Occupation measures for dynamical systems & control (Milan Korda)
    • 3:00 PM
      Coffee break
    • 12
      Trajectory recovery (atomic approximation, Christoffel-Darboux Kernel
    • 13
      The region of attraction, invariant sets and extreme values (Milan Korda)
    • 10:30 AM
      Coffee break
    • 14
      Extensions: Partial differential equations, complexity reduction (sparsity, state-space partition) (Milan Korda)
    • 12:00 PM
      Lunch break
    • 15
      Exercises (Milan Korda)