randn('state',0);
n = 10;
N = 100;
Strue = sprandsym(n,0.5,0.01,1);
nnz_true = sum(Strue(:)>1e-4);
R = inv(full(Strue));
y_sample = sqrtm(R)*randn(n,N);
Y = cov(y_sample');
Nlambda = 20;
lambda = logspace(-2, 3, Nlambda);
nnz = zeros(1,Nlambda);
for i=1:Nlambda
disp(['i = ' num2str(i) ', lambda(i) = ' num2str(lambda(i))]);
cvx_begin sdp
variable S(n,n) symmetric
maximize log_det(S) - trace(S*Y) - lambda(i)*sum(sum(abs(S)))
S >= 0
cvx_end
nnz(i) = sum(S(:)>1e-4);
end
figure;
semilogx(lambda, nnz);
hold on;
semilogx(lambda, nnz_true*ones(1,Nlambda),'r');
xlabel('\lambda');
legend('nonzeros in S', 'nonzeros in R^{-1}');
i = 1, lambda(i) = 0.01
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.125e+00 3.355e-01 1.255e-07 | Solved
1/ 1 | 5.487e-03 2.254e-06 9.933e-08 | Solved
0/ 1 | 3.910e-04 1.089e-07 9.895e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -31.3012
i = 2, lambda(i) = 0.01833
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.130e+00 3.370e-01 7.234e-08 | Solved
1/ 1 | 5.818e-03 2.430e-06 2.271e-08 | Solved
1/ 1 | 4.145e-04 3.492e-08 2.271e-08 | Solved
0/ 1 | 2.961e-05 2.277e-08 2.270e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -31.3504
i = 3, lambda(i) = 0.033598
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.138e+00 3.396e-01 4.734e-08 | Solved
1/ 1 | 6.455e-03 3.080e-06 1.118e-07 | Solved
0/ 1 | 4.597e-04 1.269e-07 1.122e-07 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -31.4368
i = 4, lambda(i) = 0.061585
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.151e+00 3.441e-01 4.352e-08 | Solved
1/ 1 | 7.595e-03 4.110e-06 6.682e-09 | Solved
1/ 1 | 5.405e-04 2.740e-08 6.681e-09 | Solved
0/ 1 | 3.862e-05 6.836e-09 6.725e-09 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -31.5845
i = 5, lambda(i) = 0.11288
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.174e+00 3.515e-01 6.697e-08 | Solved
1/ 1 | 9.467e-03 6.486e-06 1.061e-07 | Solved
1/ 1 | 6.733e-04 1.377e-07 1.058e-07 | Solved
0/ 1 | 4.811e-05 1.060e-07 1.059e-07 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -31.8272
i = 6, lambda(i) = 0.20691
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.209e+00 3.633e-01 6.798e-09 | Solved
1/ 1 | 1.248e-02 1.109e-05 1.180e-08 | Solved
1/ 1 | 8.860e-04 6.712e-08 1.195e-08 | Solved
0/ 1 | 6.331e-05 1.209e-08 1.176e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -32.2113
i = 7, lambda(i) = 0.37927
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.264e+00 3.818e-01 5.682e-08 | Solved
1/ 1 | 1.727e-02 2.127e-05 5.134e-08 | Solved
1/ 1 | 1.223e-03 1.616e-07 5.588e-08 | Solved
0/ 1 | 8.742e-05 5.606e-08 5.546e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -32.8011
i = 8, lambda(i) = 0.69519
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.344e+00 4.098e-01 1.613e-07 | Solved
1/ 1 | 2.464e-02 4.333e-05 1.018e-07 | Solved
1/ 1 | 1.740e-03 3.169e-07 1.025e-07 | Solved
0/ 1 | 1.244e-04 1.033e-07 1.021e-07 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -33.6657
i = 9, lambda(i) = 1.2743
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.454e+00 4.503e-01 3.628e-08 | Solved
1/ 1 | 3.573e-02 9.096e-05 2.945e-08 | Solved
1/ 1 | 2.510e-03 4.726e-07 3.051e-08 | Solved
0/ 1 | 1.795e-04 3.355e-08 3.081e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -34.876
i = 10, lambda(i) = 2.3357
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.601e+00 5.072e-01 2.229e-08 | Solved
1/ 1 | 5.185e-02 1.915e-04 1.271e-07 | Solved
1/ 1 | 3.612e-03 1.106e-06 1.789e-07 | Solved
0/ 1 | 2.550e-04 2.526e-08 2.071e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -36.4981
i = 11, lambda(i) = 4.2813
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 2.786e+00 5.839e-01 1.381e-08 | Solved
1/ 1 | 7.449e-02 3.953e-04 1.529e-08 | Solved
1/ 1 | 5.131e-03 1.861e-06 1.158e-08 | Solved
0/ 1 | 3.671e-04 1.401e-08 1.178e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -38.5618
i = 12, lambda(i) = 7.8476
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 3.011e+00 6.842e-01 7.251e-09 | Solved
1/ 1 | 1.053e-01 7.906e-04 3.331e-09 | Solved
1/ 1 | 7.155e-03 3.630e-06 5.011e-09 | Solved
1/ 1 | 5.127e-04 2.168e-08 5.055e-09 | Solved
0/ 1 | 3.661e-05 2.136e-09 5.052e-09 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -41.092
i = 13, lambda(i) = 14.3845
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 3.273e+00 8.124e-01 1.292e-10 | Solved
1/ 1 | 1.463e-01 1.528e-03 9.899e-08 | Solved
1/ 1 | 9.745e-03 6.735e-06 7.207e-08 | Solved
0/ 1 | 7.002e-04 3.429e-08 6.008e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -44.1061
i = 14, lambda(i) = 26.3665
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 3.584e+00 9.786e-01 1.254e-09 | Solved
1/ 1 | 2.014e-01 2.896e-03 1.615e-08 | Solved
1/ 1 | 1.300e-02 1.199e-05 1.282e-10 | Solved
1/ 1 | 9.410e-04 6.470e-08 7.387e-09 | Solved
0/ 0 | 0.000e+00 0.000e+00 0.000e+00 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -47.7294
i = 15, lambda(i) = 48.3293
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 3.940e+00 1.190e+00 2.178e-08 | Solved
1/ 1 | 2.737e-01 5.358e-03 4.829e-10 | Solved
1/ 1 | 1.705e-02 2.062e-05 1.092e-09 | Solved
1/ 1 | 1.227e-03 1.110e-07 1.033e-09 | Solved
1/ 1 | 8.758e-05 1.404e-09 1.043e-09 | Solved
0/ 1 | 6.256e-06 1.075e-09 1.038e-09 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -51.9812
i = 16, lambda(i) = 88.5867
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 4.335e+00 1.448e+00 9.696e-09 | Solved
1/ 1 | 3.644e-01 9.520e-03 3.968e-08 | Solved
1/ 1 | 2.159e-02 3.315e-05 3.598e-08 | Solved
1/ 1 | 1.560e-03 1.707e-07 4.037e-08 | Solved
0/ 1 | 1.112e-04 6.868e-10 3.981e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -56.7913
i = 17, lambda(i) = 162.3777
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 4.755e+00 1.752e+00 4.613e-09 | Solved
1/ 1 | 4.738e-01 1.613e-02 2.491e-08 | Solved
1/ 1 | 2.631e-02 4.924e-05 5.624e-08 | Solved
0/ 1 | 1.856e-03 2.248e-07 1.052e-06 | Inaccurate/Solved
0/ 1 | 9.017e-05 1.853e-11 1.060e-07 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -62.0441
i = 18, lambda(i) = 297.6351
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 5.190e+00 2.097e+00 1.453e-07 | Solved
1/ 1 | 6.002e-01 2.595e-02 1.851e-10 | Solved
1/ 1 | 3.072e-02 6.695e-05 2.359e-09 | Solved
1/ 1 | 2.226e-03 3.612e-07 2.405e-09 | Solved
0/ 1 | 1.589e-04 6.270e-10 2.401e-09 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -67.6118
i = 19, lambda(i) = 545.5595
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 5.629e+00 2.479e+00 1.728e-06 | Solved
1/ 1 | 7.412e-01 3.970e-02 1.001e-07 | Solved
1/ 1 | 3.437e-02 8.404e-05 1.021e-07 | Solved
1/ 1 | 2.497e-03 4.346e-07 4.581e-08 | Solved
0/ 1 | 1.646e-04 1.906e-09 1.348e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -73.3839
i = 20, lambda(i) = 1000
Successive approximation method to be employed.
sedumi will be called several times to refine the solution.
Original size: 502 variables, 279 equality constraints
1 exponentials add 8 variables, 5 equality constraints
-----------------------------------------------------------------
Cones | Errors |
Mov/Act | Centering Exp cone Poly cone | Status
--------+---------------------------------+---------
1/ 1 | 6.068e+00 2.891e+00 8.466e-07 | Solved
1/ 1 | 8.947e-01 5.802e-02 6.623e-07 | Solved
1/ 1 | 3.671e-02 9.585e-05 1.067e-07 | Solved
1/ 1 | 2.682e-03 5.072e-07 8.768e-08 | Solved
0/ 1 | 2.031e-04 1.601e-09 9.269e-08 | Solved
-----------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -79.2801