# Section 4.5.4: Frobenius norm diagonal scaling (GP)

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    % 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: 82 variables, 32 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  4.018e-04  0.000e+00 | Solved
      2/  9 | 1.198e-03  7.548e-08  0.000e+00 | Solved
      0/  0 | 2.616e-05  0.000e+00  0.000e+00 | Solved
    -----------------------------------------------------------------
    Status: Solved
    Optimal value (cvx_optval): +3.25231

    The matrix D that minimizes ||DMD^{-1}||_F is:
        1.0885         0         0         0
             0    0.9138         0         0
             0         0    0.9754         0
             0         0         0    1.6294

    The minimium Frobenius norm achieved is:
        3.2523

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