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