Course on polynomial and LMI optimization

Contributed by: 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

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.


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.


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).