Thank you both very much for your answer. I was already aware of Matlink but I was thinking of finding a way of doing it from mathematica without calling external programs. I like the idea of trying to use FindMinimum. If I can find a way of making it work I will share my result, I think it could be something handy for lots of people.
Cheers
FindMinimum can handle vector variables only for unconstrained problems, unlike CVX which can take vector and array variables.
One possibility is to reformulate as an explicit optimization with constraints on variables (that is, not in terms of matrix inequalities). The link below covers the basics of how this could be done (maybe it's well known to you already). Then you can just call FindMinimum giving the objective and constraints in a list.
http://en.wikipedia.org/wiki/Semidefinite_programming
Also there may be some useful ideas at the various links below.
http://forums.wolfram.com/mathgroup/archive/2010/Dec/msg00540.html
http://stackoverflow.com/questions/5151178/quadratic-programming-in-mathematica
http://mathematica-bits.blogspot.com/2011/03/semidefinite-programming-in-mathematica.html
MATLAB tools, including CVX, can be called from Mathematica using MATLINK, which has been discussed on the Forum. If you do a search on MATLINK in the forum, you'll find the discussion.
