The point of this work is given in the short paper "Alternatives with stronger convergence than coordinate-descent iterative LMI algorithms" http://arxiv.org/abs/1110.2615
In short, the point is to illustrate that instead of developing or using what can be described as coordinate-descent iterative LMI algorithms, which are not guaranteed to converge even to locally optimal solutions, there exist many methods with better convergence properties (in theory and/or in practice).
This should be of interest for a sizeable part of the systems and control community, considering the large number of research still being made on developping this kind of algorithms which are not supported by a (strong) convergence analysis.
Note that there exist some few iterative LMI algorithms that have such proof of convergence, and these do not rely (at all or only) on a coordinate descent principle, see for instance http://www.kuleuven.be/optec/software/BMIsolver (copy on http://arxiv.org/abs/1109.3320).
The problem considered is the same as in "Hinf Positive Filtering for Positive Linear Discrete-Time Systems: An Augmentation Approach": http://dx.doi.org/10.1109/TAC.2010.2053471
The issue with the technique proposed in that paper is that it is not guaranteed to converge even to locally optimal solutions.
In theory, there are optimization methods with better convergence properties that would therefore be more adequate.
In practice, there are many optimization methods quickly available and easy to use which will lead to solutions with better performance levels.
Emile Simon (2020). A direct search method instead of a coordinate descent iterative LMI algorithm (https://www.mathworks.com/matlabcentral/fileexchange/33219-a-direct-search-method-instead-of-a-coordinate-descent-iterative-lmi-algorithm), MATLAB Central File Exchange. Retrieved .
For more details about restarting Nelder-Mead (e.g. fminsearch) to improve its convergence, see the description and file on http://www.mathworks.com/matlabcentral/fileexchange/33328
The first version of this program has a small implementation problem. The second version will be available shortly, with more comments.
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