Minimal phase spectral factorization
b = [2; 0.2; -0.3];
n = length( b );
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{:} ]';
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 SDPT3: 6 variables, 3 equality constraints
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num. of constraints = 3
dim. of sdp var = 3, num. of sdp blk = 1
*******************************************************************
SDPT3: Infeasible path-following algorithms
*******************************************************************
version predcorr gam expon scale_data
HKM 1 0.000 1 0
it pstep dstep pinfeas dinfeas gap mean(obj) cputime
-------------------------------------------------------------------
0|0.000|0.000|2.6e+00|1.3e+00|3.2e+01| 4.941343e+00| 0:0:00| chol 1 1
1|1.000|1.000|4.6e-07|5.4e-02|3.5e+00|-4.793715e-01| 0:0:00| chol 1 1
2|1.000|0.906|6.8e-08|9.9e-03|3.9e-01| 2.637395e-01| 0:0:00| chol 1 1
3|0.967|1.000|1.2e-08|5.4e-04|1.1e-02| 1.268420e-01| 0:0:00| chol 1 1
4|0.945|1.000|3.7e-08|5.4e-05|4.8e-04| 1.229281e-01| 0:0:00| chol 1 1
5|0.976|1.000|1.2e-08|5.4e-06|4.7e-05| 1.227472e-01| 0:0:00| chol 1 1
6|0.932|1.000|1.4e-09|2.5e-09|2.2e-06| 1.227328e-01| 0:0:00| chol 1 1
7|1.000|1.000|1.9e-10|2.8e-10|2.2e-07| 1.227326e-01| 0:0:00| chol 2 2
8|1.000|1.000|2.3e-11|3.8e-11|9.6e-09| 1.227326e-01| 0:0:00|
stop: max(relative gap, infeasibilities) < 1.49e-08
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number of iterations = 8
primal objective value = 1.22732571e-01
dual objective value = 1.22732561e-01
gap := trace(XZ) = 9.60e-09
relative gap = 7.71e-09
actual relative gap = 7.63e-09
rel. primal infeas = 2.26e-11
rel. dual infeas = 3.83e-11
norm(X), norm(y), norm(Z) = 2.0e+00, 7.6e-01, 2.4e+00
norm(A), norm(b), norm(C) = 3.1e+00, 3.0e+00, 3.2e+00
Total CPU time (secs) = 0.1
CPU time per iteration = 0.0
termination code = 0
DIMACS: 2.3e-11 0.0e+00 4.1e-11 0.0e+00 7.6e-09 7.7e-09
-------------------------------------------------------------------
------------------------------------------------------------
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):
1.1049e-09
The optimal weighted trace:
0.1227