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<title>Closest Toeplitz SDP search.</title>
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<h1>Closest Toeplitz SDP search.</h1>
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<pre class="codeinput">
<span class="comment">% This script finds a Toeplitz Hermitian PSD matrix that is closest to a</span>
<span class="comment">% given Hermitian matrix, as measured by the Frobenius norm. That is, for</span>
<span class="comment">% a given matrix P, it solves:</span>
<span class="comment">% minimize || Z - P ||_F</span>
<span class="comment">% subject to Z &gt;= 0</span>
<span class="comment">%</span>
<span class="comment">% Adapted from an example provided in the SeDuMi documentation. Notice</span>
<span class="comment">% the use of SDP mode to simplify the semidefinite constraint.</span>
<span class="comment">% The data. P is Hermitian, but is neither Toeplitz nor PSD.</span>
P = [ 4, 1+2*j, 3-j ; <span class="keyword">...</span>
1-2*j, 3.5, 0.8+2.3*j ; <span class="keyword">...</span>
3+j, 0.8-2.3*j, 4 ];
<span class="comment">% Construct and solve the model</span>
n = size( P, 1 );
cvx_begin <span class="string">sdp</span>
variable <span class="string">Z(n,n)</span> <span class="string">hermitian</span> <span class="string">toeplitz</span>
dual <span class="string">variable</span> <span class="string">Q</span>
minimize( norm( Z - P, <span class="string">'fro'</span> ) )
Z &gt;= 0 : Q;
cvx_end
<span class="comment">% Display resuls</span>
disp( <span class="string">'The original matrix, P: '</span> );
disp( P )
disp( <span class="string">'The optimal point, Z:'</span> );
disp( Z )
disp( <span class="string">'The optimal dual variable, Q:'</span> );
disp( Q )
disp( <span class="string">'min( eig( Z ) ), min( eig( Q ) ) (both should be nonnegative, or close):'</span> );
disp( sprintf( <span class="string">' %g %g\n'</span>, min( eig( Z ) ), min( eig( Q ) ) ) );
disp( <span class="string">'The optimal value, || Z - P ||_F:'</span> );
disp( norm( Z - P, <span class="string">'fro'</span> ) );
disp( <span class="string">'Complementary slackness: Z * Q, should be near zero:'</span> );
disp( Z * Q )
</pre>
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<pre class="codeoutput">
Calling SDPT3: 20 variables, 6 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------
num. of constraints = 6
dim. of sdp var = 6, num. of sdp blk = 1
dim. of socp var = 11, num. of socp 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 prim-obj dual-obj cputime
-------------------------------------------------------------------
0|0.000|0.000|1.8e+01|2.7e+00|7.4e+02| 1.260000e+02 0.000000e+00| 0:0:00| chol 1 1
1|0.332|1.000|1.2e+01|3.3e-02|4.7e+02| 9.616817e+01 -4.813672e+01| 0:0:00| chol 1 1
2|1.000|1.000|2.4e-07|3.3e-03|3.1e+01| 4.813335e+00 -2.613716e+01| 0:0:00| chol 1 1
3|1.000|0.793|9.3e-08|9.5e-04|7.1e+00| 2.763897e+00 -4.276057e+00| 0:0:00| chol 1 1
4|0.780|1.000|2.8e-08|3.3e-05|1.9e+00|-9.411558e-01 -2.847823e+00| 0:0:00| chol 1 1
5|0.979|0.962|8.9e-10|4.5e-06|6.6e-02|-1.414992e+00 -1.480982e+00| 0:0:00| chol 1 1
6|0.983|0.989|1.3e-10|3.8e-07|9.5e-04|-1.450304e+00 -1.451248e+00| 0:0:00| chol 1 1
7|0.939|0.983|9.0e-10|3.9e-08|3.9e-05|-1.450788e+00 -1.450827e+00| 0:0:00| chol 1 1
8|0.932|0.984|2.0e-09|6.7e-10|1.8e-06|-1.450803e+00 -1.450804e+00| 0:0:00| chol 1 1
9|0.981|0.990|1.0e-10|6.6e-11|5.7e-08|-1.450803e+00 -1.450804e+00| 0:0:00|
stop: max(relative gap, infeasibilities) &lt; 1.49e-08
-------------------------------------------------------------------
number of iterations = 9
primal objective value = -1.45080348e+00
dual objective value = -1.45080354e+00
gap := trace(XZ) = 5.69e-08
relative gap = 1.46e-08
actual relative gap = 1.45e-08
rel. primal infeas = 1.05e-10
rel. dual infeas = 6.61e-11
norm(X), norm(y), norm(Z) = 1.9e+00, 3.2e+00, 6.8e+00
norm(A), norm(b), norm(C) = 5.5e+00, 2.0e+00, 8.0e+00
Total CPU time (secs) = 0.15
CPU time per iteration = 0.02
termination code = 0
DIMACS: 1.0e-10 0.0e+00 1.0e-10 0.0e+00 1.4e-08 1.5e-08
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +1.4508
The original matrix, P:
4.0000 1.0000 + 2.0000i 3.0000 - 1.0000i
1.0000 - 2.0000i 3.5000 0.8000 + 2.3000i
3.0000 + 1.0000i 0.8000 - 2.3000i 4.0000
The optimal point, Z:
4.2827 0.8079 + 1.7342i 2.5574 - 0.7938i
0.8079 - 1.7342i 4.2827 0.8079 + 1.7342i
2.5574 + 0.7938i 0.8079 - 1.7342i 4.2827
The optimal dual variable, Q:
0.3366 -0.0635 - 0.2866i -0.3051 + 0.1422i
-0.0635 + 0.2866i 0.2561 -0.0635 - 0.2866i
-0.3051 - 0.1422i -0.0635 + 0.2866i 0.3366
min( eig( Z ) ), min( eig( Q ) ) (both should be nonnegative, or close):
1.08289e-08 2.17537e-09
The optimal value, || Z - P ||_F:
1.4508
Complementary slackness: Z * Q, should be near zero:
1.0e-05 *
0.0915 - 0.1253i -0.1237 - 0.0530i -0.0286 + 0.1516i
0.0448 - 0.2020i -0.1790 - 0.0000i 0.0448 + 0.2020i
-0.0286 - 0.1516i -0.1237 + 0.0530i 0.0915 + 0.1253i
</pre>
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