Minimal phase spectral factorization

% A PSD matrix is found which minimizes a weighted trace while obtaining
% fixed sums along the diagonals. Notice the use of a FOR loop to access
% the diagonals of X. A later version of CVX will eliminate the need for
% this by allowing the use of the SPDIAGS function in side models.
% Nevertheless, this example provides an illustration of the use of
% standard Matlab control statements to build models.
%
% Adapted from an example provided in the SeDuMi documentation.

% Generate data
b = [2; 0.2; -0.3];
n = length( b );

% Create and solve model
cvx_begin sdp
    variable X( n, n ) symmetric
    dual variable y{n}
    minimize( ( n - 1 : -1 : 0 ) * diag( X ) );
    for k = 1 : n,
        sum( diag( X, k - 1 ) ) == b( k ) : y{k};
    end
    X >= 0;
cvx_end
y = [ y{:} ]';

% Display resuls
disp( 'The optimal point, X:' );
disp( X )
disp( 'The diagonal sums:' );
disp( sum( spdiags( X, 0:n-1 ), 1 ) );
disp( 'min( eig( X ) ) (should be nonnegative):' );
disp( min( eig( X ) ) )
disp( 'The optimal weighted trace:' );
disp( ( n - 1 : -1 : 0 ) * diag( X ) );
 
Calling SeDuMi: 6 variables (0 free), 3 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 3, order n = 4, dim = 10, blocks = 2
nnz(A) = 6 + 0, nnz(ADA) = 9, nnz(L) = 6
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            1.69E+001 0.000
  1 : -2.44E-001 3.91E+000 0.000 0.2312 0.9000 0.9000   1.39  1  1  2.0E+000
  2 :  1.86E-001 3.22E-001 0.000 0.0824 0.9900 0.9900   1.31  1  1  4.0E-001
  3 :  1.23E-001 1.12E-004 0.000 0.0003 0.9999 0.9999   0.99  1  1  2.6E-004
  4 :  1.23E-001 5.18E-006 0.000 0.0463 0.9900 0.9900   1.00  1  1  1.2E-005
  5 :  1.23E-001 2.36E-007 0.000 0.0455 0.9900 0.9900   1.00  1  1  6.8E-007
  6 :  1.23E-001 1.90E-008 0.327 0.0806 0.9900 0.9900   1.00  2  2  5.5E-008
  7 :  1.23E-001 8.16E-010 0.000 0.0429 0.9906 0.9900   0.99  2  2  2.2E-009

iter seconds digits       c*x               b*y
  7      0.0   8.5  1.2273256559e-001  1.2273256524e-001
|Ax-b| =  1.1e-010, [Ay-c]_+ =  1.4E-010, |x|= 2.0e+000, |y|= 7.6e-001

Detailed timing (sec)
   Pre          IPM          Post
1.001E-002    1.001E-002    0.000E+000    
Max-norms: ||b||=2, ||c|| = 2,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 9.99993.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.122733
The optimal point, X:
    0.0468   -0.0369   -0.3000
   -0.0369    0.0292    0.2369
   -0.3000    0.2369    1.9240

The diagonal sums:
    2.0000    0.2000   -0.3000

min( eig( X ) ) (should be nonnegative):
  4.4503e-011

The optimal weighted trace:
    0.1227