Seminar Details
2010-11-16 (15.00) : Optimization & Control: Solving BMI problems via LMI coordinate descent
At Euler building
Organized by Mathematical Engineering
Speaker :
Emile SIMON (CESAME)
Abstract :
In this talk we will discuss a method to solve Bilinear Matrix
Inequalities (BMIs), in particular in their original field of optimal
control. Starting from an initial solution, the idea used is to fix one
side of the BMIs and solve the resulting LMIs problem, then fix this new
side, free the other side of the BMIs, solve the new LMIs problem and
iterate between these two steps. This is in fact a coordinate descent
algorithm. The motivation is to reach a local optimum in polynomial time
(difficult because BMIs are in general non-convex and N-P hard). This
method is mostly overlooked because it is not guaranteed to reach the
global optimum but also not even local optimae in general. Indeed we will
see that the first algorithm proposed in the literature implementing this
technique, fixing either all of the Lyapunov matrix(ces) or the control
parameter of the optimal control problem (so-called V-K iteration), cannot
work and why.
The contributions of the current work are the following. First, the
problem and difficulties are outlined with a short survey of relevant
papers and useful results. Then, the use of a particular change of
variables (thus of coordinates) is made so that the algorithm will improve
the initial suboptimal solutions (for fixed-order output-feedback control
design for LTI systems). Finally, an important improvement is
proposed that probably ensures convergence towards local optimality.
Although the exact convergence conditions are still under investigation,
the algorithm (that could be interpreted as a BMI interior point method)
performs well in practice which will be shown by means of an example.