% Boyd & Vandenberghe "Convex Optimization"
% Joelle Skaf - 01/29/06
% Updated to use GP mode by Almir Mutapcic 02/08/06
%
% Given a square matrix M, the goal is to find a vector (with dii > 0)
% such that ||DMD^{-1}||_F is minimized, where D = diag(d).
% The problem can be cast as an unconstrained geometric program:
%           minimize sqrt( sum_{i,j=1}^{n} Mij^2*di^2/dj^2 )
%

rs = randn( 'state' );
randn( 'state', 0 );

% matrix size (M is an n-by-n matrix)
n = 4;
M = randn(n,n);

% formulating the problem as a GP
cvx_begin gp
  variable d(n)
  minimize( sqrt( sum( sum( diag(d.^2)*(M.^2)*diag(d.^-2) ) ) ) )
  % Alternate formulation: norm( diag(d)*abs(M)*diag(1./d), 'fro' )
cvx_end

% displaying results
D = diag(d);
disp('The matrix D that minimizes ||DMD^{-1}||_F is: ');
disp(D);
disp('The minimium Frobenius norm achieved is: ');
disp(norm(D*M*inv(D),'fro'));
disp('while the Frobunius norm of the original matrix M is: ');
disp(norm(M,'fro'));
 
Successive approximation method to be employed.
   SDPT3 will be called several times to refine the solution.
   Original size: 55 variables, 34 equality constraints
   16 exponentials add 128 variables, 80 equality constraints
-----------------------------------------------------------------
 Cones  |             Errors              |
Mov/Act | Centering  Exp cone   Poly cone | Status
--------+---------------------------------+---------
 16/ 16 | 8.000e+00  1.276e+01  0.000e+00 | Solved
 16/ 16 | 1.590e+00  1.546e-01  0.000e+00 | Solved
 15/ 15 | 7.950e-02  4.017e-04  0.000e+00 | Solved
  0/  3 | 1.053e-03  6.705e-09  0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +3.25231
 
The matrix D that minimizes ||DMD^{-1}||_F is: 
    1.0864         0         0         0
         0    0.9120         0         0
         0         0    0.9735         0
         0         0         0    1.6263

The minimium Frobenius norm achieved is: 
    3.2523

while the Frobunius norm of the original matrix M is: 
    3.6126