minimizing condition number of a matrix

[mar sag]

Dear Johan, 

I'm starting to use Yalmip, and I'd like to implement an example taken from the book of Prof. Boyd "Linear Matrix Inequalities in System and Control Theory". Specifically, I'd like to implement an example of what illustrated in Sec. 3.1 (pag 37) of the book, about the scaling of a rectangular matrix M via diagonal, PSD matrices. 

I just would like to know whether I'm doing anything wrong, because according to the syntax of Yalmip, it should be ok. 

since I want to minimize gamma^2, I defined a variable gam2 > 1, which should be equivalent. For the rest I´m using the same nomenclature of the book. Am I doing anything weird here? Thanks a lot, 

Marco

my code is

yalmip('clear');

% defining the matrix to scale

M = [100 23 2; 40 1 8; 6 .0004 1; 5000 2 1];

% check the condition number

cond(M);

% define the dimensions of the matrices P and Q

dim = size(M);

p = dim(1);

q = dim(2);

% define in Yalmip the matrices P and Q, theymust be PSD matrices

P = sdpvar(p,p,'diagonal');

Q = sdpvar(q,q,'diagonal');

% define the scalar gamma^2. Since it is a condition number, it has to be equal or greater than 1

gam2 = sdpvar(1,1);

% assign the LMI conditions: P, Q PSD, and Q<=M' P M <= gamma^2*Q

F = [P>0, Q>0, gam2>1,Q<=M'*P*M,M'*P*M<=gam2*Q];

% solve

optimize(F,gam2)

% check the values

value(gam2)

value(Q)

value(P)

% extract the matrices L and R

L = chol(value(P),'lower');

R = chol(Qi,'lower');

% check the new condition number

cond(L*M*R)

what I get is

ans = 

    solvertime: 0

          info: 'No suitable solver'

       problem: -2

    yalmiptime: 1.222

any suggestion? thanks!

[Johan Löfberg]

You multiply gam2 and Q, hence it is not an LMI, and thus you most likely don't have any solver.

However, the problem is quasi-convex and easily solved using bisection in gam2. See the example decay-rate minimization on the wiki, and the lazy-mans command bisection.

BTW, you cannot use strict inequalities. YALMIP is screaming at you about this.

[mar sag]

Dear Johan, 

thanks for your reply. So, formally not all the GEVPs are LMIs? or my problem is not a GEVP?

P.S. Yes, Yalmip was screaming about strict inequalities, so I changed them :)

Kind Regards,

Marco

[Johan Löfberg]

No GEVP is an LMI, by definition. LMI stands for LINEAR matrix inequality. In a GEVP, you have a bilinear product. They can easily be solved through a small number of LMIs though, by bisection as illustrated.

[mar sag]

Thanks! One last question. Is there an updated Yalmip user manual I can download and print? or do I have to use the website? I found a user manual written by you, but I think it is obsolete, as it contains still commands like "addlmi", which are now recognized by yalmip as obsolete. Any new version? thanks again, 

Regards,

Marco

[Johan Löfberg]

No, Wiki is the only manual

[mar sag]

Ok, many thanks!