[Opt-Net] Course on polynomial and LMI optimization in Prague, February 2015

[Didier Henrion]

Course on polynomial and LMI optimization with applications in control
by Didier Henrion, LAAS-CNRS, Toulouse, France
and Czech Technical University in Prague, Czech Republic.

http://homepages.laas.fr/henrion/courses/lmi15

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).
It consists of six two-hour lectures, given on Monday 16, Thursday 19 and
Monday 23 February, 2015, from 10am to noon and from 2pm to 4pm.

Registration:
There is no admission fee, students and reseachers from external 
institutions
are particularly welcome, but please send an e-mail to <henrion at laas.fr> 
to register.

Target audience:
This is a course for graduate students or researchers with some background
in linear algebra, convex optimization and linear control systems.

Outline:
Many problems of systems control theory boil down to solving polynomial 
equations,
polynomial inequalities or polyomial differential equations. Recent 
advances in convex
optimization and real algebraic geometry can be combined to generate 
approximate
solutions in floating point arithmetic.
In the first part of the course we describe semidefinite programming 
(SDP) as an extension
of linear programming (LP) to the cone of positive semidefinite 
matrices. We investigate
the geometry of spectrahedra, convex sets defined by linear matrix 
inequalities (LMIs)
or affine sections of the SDP cone. We also introduce spectrahedral 
shadows, or lifted
LMIs, obtained by projecting affine sections of the SDP cones. Then we 
review existing
numerical algorithms for solving SDP problems.
In the second part of the course we describe several recent applications 
of SDP. First, we
explain how to solve polynomial optimization problems, where a real 
multivariate polynomial
must be optimized over a (possibly nonconvex) basic semialgebraic set. 
Second, we
extend these techniques to ordinary differential equations (ODEs) with 
polynomial
dynamics, and the problem of trajectory optimization (analysis of 
stability or performance
of solutions of ODEs). Third, we conclude this part with applications to 
optimal control
(design of a trajectory optimal w.r.t. a given functional).

-- 

Didier Henrion
http://homepages.laas.fr/henrion