##
Course on LMI optimization with applications in control

(Didier Henrion, LAAS-CNRS, Toulouse, France and FEL-CVUT, Prague,
Czech Republic)

### Czech Technical University,
Prague, Czech Republic - March 2011

## Venue and dates

The course is given at the Charles Square campus of the
Czech Technical University, in the historical center of
Prague (Karlovo Namesti 13, 12135
Praha 2) during the first two weeks of March 2011.
It consists of six two-hour lectures, given on Tuesday 1 March,
Thursday 3 March and Tuesday 8 March, from 10am to noon,
and from 2pm to 4pm.

## Description

This is a course for graduate students or researchers
with a background in linear control
systems, linear algebra and convex optimization.
The focus is on semidefinite programming (SDP), or
optimization over linear matrix inequalities (LMIs), an extension of
linear programming to the cone of positive semidefinite matrices. Since
the 1990s, LMI methods have found numerous applications mostly in
combinatorial optimization, systems control and signal processing.

## Outline

The course starts with fundamental
mathematical features of linear matrix inequalities:
Part 0: general introduction, course outline and material

Part 1: technical background on linear algebra, numerical
methods, cones, duality and convexity

Part 2: what is an LMI ? history, connection with semidefinite
programming (SDP)

Part 3: LMI duality, alternatives, Farkas lemma, S-procedure

More recent material covers the applications of LMIs
in polynomial optimization:

Part 4: convex LMI modelling,
classification of convex semialgebraic sets that can
be represented with LMIs, lift and project techniques

Part 5: nonconvex LMI modelling, BMI,
Shor's relaxation, polynomial moments and sum-of-squares,
Lasserre's hierarchy of LMI relaxations for non-convex polynomial
optimization

We also briefly survey recent developments in semidefinite solvers and
software packages:

Part 6: LMI solvers, basics of interior-point algorithms,
latest achievements in software and solvers for LMIs and BMIs

The end of the course focuses on the use of measures
for static polynomial optimisation problems and occupation
measures for differential equations and related optimal
control problems:

Part 7<: Measures, occupation measures
and control problems

This last part surveys most of the material covered in Parts 5,
but through the lense of measure theory and the related
generalized problem of moments. Moreover, it extends
these ideas to differential equations and optimal control framework,
and it describes several research directions along these lines.

## Homeworks and exam

Homeworks are handed out during the course.
A written examination can be organized.

Last updated on 24 February 2011.