Course on polynomial and LMI optimization with applications in control
(Didier Henrion, LAAS-CNRS, Toulouse, France and FEL-CVUT, Prague, Czech Republic)

Czech Technical University, Prague, Czech Republic - 18-20 April 2016

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 third week of April 2016. It consists of six two-hour lectures, given on Monday 18, Tuesday 19 and Wednesday 20 April, from 10am to noon and from 2pm to 4pm.

The course is given in room E-24, ground floor on your left hand side when entering building E. Please refer to these maps for instructions to reach this building. The campus entrance is restricted to badge holders, so please ask the doorperson to let you in through the electronic gates.


The course is primarily aimed at students from the Czech Technical University in Prague, yet external participants are welcome. There is no registration fee. Please note that the Czech Technical University will not provide assistance regarding traveling and accomodation in Prague.


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

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).


The course closely follows the lecture notes:

  • D. Henrion. Optimization on linear matrix inequalities for polynomial systems control. International Summer School of Automatic Control, Grenoble, France, September 2014.

    Homeworks and exam

    Homeworks are given during the course. A written examination can be organized. Certificates of attendance can be provided on request.

    Last updated on 19 January 2016.