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### Didier Henrion - Course on polynomial and LMI optimization in Prague

Date: 16/02/2015 10:00 - 23/02/2015 16:00

Place: The Czech Technical University -- Karlovo Namesti 13, 12135 Praha 2

Course on polynomial and LMI optimization in Prague, February 2015Contributed by: Didier Henrion, henrion@laas.fr 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/lmi15Venue 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 4pmRegistration:There is no admission fee, students and reseachers from external

institutions are particularly welcome, but please send an e-mail to

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).