Title: Optimization & Control: Solving BMI problems via LMI coordinate descent
Speaker: Emile Simon
Department of Mathematical Engineering
Universite Catholique de Louvain
Place: K.U. Leuven, Department of Computer Science
Celestijenlaan 200A, B-3001 Heverlee
Room A00.144
Date: Wednesday December 22, 2010, 13.00-14.00
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 main idea 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.
