n = 4;
fprintf(1,'Solving the upper bound SDP ...');
cvx_begin sdp
variable C1(n,n) symmetric
maximize ( C1(1,4) )
C1 >= 0;
diag(C1) == ones(n,1);
C1(1,2) >= 0.6;
C1(1,2) <= 0.9;
C1(1,3) >= 0.8;
C1(1,3) <= 0.9;
C1(2,4) >= 0.5;
C1(2,4) <= 0.7;
C1(3,4) >= -0.8;
C1(3,4) <= -0.4;
cvx_end
fprintf(1,'Done! \n');
fprintf(1,'Solving the lower bound SDP ...');
cvx_begin sdp
variable C2(n,n) symmetric
minimize ( C2(1,4) )
C2 >= 0;
diag(C2) == ones(n,1);
C2(1,2) >= 0.6;
C2(1,2) <= 0.9;
C2(1,3) >= 0.8;
C2(1,3) <= 0.9;
C2(2,4) >= 0.5;
C2(2,4) <= 0.7;
C2(3,4) >= -0.8;
C2(3,4) <= -0.4;
cvx_end
fprintf(1,'Done! \n');
disp('--------------------------------------------------------------------------------');
disp(['The minimum and maximum values of rho_14 are: ' num2str(C2(1,4)) ' and ' num2str(C1(1,4))]);
disp('with corresponding correlation matrices: ');
disp(C2)
disp(C1)
Solving the upper bound SDP ...
Calling SDPT3 4.0: 18 variables, 6 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------
num. of constraints = 6
dim. of sdp var = 4, num. of sdp blk = 1
dim. of linear var = 8
*******************************************************************
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|5.0e-01|8.7e+00|1.2e+03| 5.000000e+01 0.000000e+00| 0:0:00| chol 1 1
1|1.000|0.963|4.7e-07|4.0e-01|8.6e+01| 3.764390e+01 1.045278e-01| 0:0:00| chol 1 1
2|0.930|0.962|2.4e-06|2.4e-02|1.2e+01| 9.519382e+00 5.545477e-02| 0:0:00| chol 1 1
3|0.979|1.000|1.4e-07|9.0e-04|6.4e-01| 6.821346e-01 4.804692e-02| 0:0:00| chol 1 1
4|1.000|1.000|5.8e-08|9.0e-05|1.3e-01| 3.354711e-01 2.021144e-01| 0:0:00| chol 1 1
5|0.945|0.990|4.5e-09|9.9e-06|6.7e-03| 2.357452e-01 2.290535e-01| 0:0:00| chol 1 1
6|0.891|1.000|9.7e-09|9.0e-07|9.1e-04| 2.306737e-01 2.297716e-01| 0:0:00| chol 1 1
7|1.000|1.000|5.5e-09|9.2e-08|8.9e-05| 2.299788e-01 2.298904e-01| 0:0:00| chol 1 1
8|0.966|0.983|8.2e-10|2.6e-09|2.9e-06| 2.299114e-01 2.299086e-01| 0:0:00| chol 1 1
9|1.000|1.000|3.7e-09|1.6e-10|6.1e-07| 2.299095e-01 2.299089e-01| 0:0:00| chol 1 1
10|1.000|1.000|8.4e-12|2.5e-10|3.0e-08| 2.299091e-01 2.299091e-01| 0:0:00| chol 1 1
11|1.000|1.000|4.0e-13|1.7e-12|5.0e-10| 2.299091e-01 2.299091e-01| 0:0:00|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 11
primal objective value = 2.29909084e-01
dual objective value = 2.29909083e-01
gap := trace(XZ) = 4.98e-10
relative gap = 3.41e-10
actual relative gap = 3.35e-10
rel. primal infeas (scaled problem) = 3.97e-13
rel. dual " " " = 1.68e-12
rel. primal infeas (unscaled problem) = 0.00e+00
rel. dual " " " = 0.00e+00
norm(X), norm(y), norm(Z) = 2.8e+00, 5.7e-01, 2.8e+00
norm(A), norm(b), norm(C) = 5.5e+00, 2.0e+00, 3.8e+00
Total CPU time (secs) = 0.17
CPU time per iteration = 0.02
termination code = 0
DIMACS: 4.0e-13 0.0e+00 3.2e-12 0.0e+00 3.4e-10 3.4e-10
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.229909
Done!
Solving the lower bound SDP ...
Calling SDPT3 4.0: 18 variables, 6 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------
num. of constraints = 6
dim. of sdp var = 4, num. of sdp blk = 1
dim. of linear var = 8
*******************************************************************
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|5.0e-01|8.7e+00|1.2e+03| 5.000000e+01 0.000000e+00| 0:0:00| chol 1 1
1|1.000|0.965|4.6e-07|3.9e-01|8.5e+01| 3.764990e+01 1.075756e-01| 0:0:00| chol 1 1
2|0.927|1.000|2.3e-06|9.0e-03|9.9e+00| 9.200175e+00 1.305318e-01| 0:0:00| chol 1 1
3|0.897|1.000|2.9e-07|9.0e-04|1.0e+00| 1.182476e+00 1.678502e-01| 0:0:00| chol 1 1
4|1.000|0.662|4.2e-08|3.6e-04|6.2e-01| 9.449252e-01 3.321156e-01| 0:0:00| chol 1 1
5|0.933|1.000|4.1e-09|9.0e-06|4.2e-02| 4.233944e-01 3.816576e-01| 0:0:00| chol 1 1
6|0.963|1.000|1.1e-09|9.0e-07|4.5e-03| 3.962889e-01 3.917716e-01| 0:0:00| chol 1 1
7|0.960|0.992|9.3e-10|9.7e-08|2.6e-04| 3.930398e-01 3.927797e-01| 0:0:00| chol 1 1
8|0.966|0.985|6.9e-10|1.1e-08|8.3e-06| 3.928279e-01 3.928197e-01| 0:0:00| chol 1 1
9|1.000|1.000|2.2e-09|1.4e-10|1.6e-06| 3.928215e-01 3.928199e-01| 0:0:00| chol 1 1
10|1.000|1.000|2.1e-11|2.1e-10|2.3e-08| 3.928203e-01 3.928203e-01| 0:0:00|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 10
primal objective value = 3.92820340e-01
dual objective value = 3.92820317e-01
gap := trace(XZ) = 2.32e-08
relative gap = 1.30e-08
actual relative gap = 1.25e-08
rel. primal infeas (scaled problem) = 2.14e-11
rel. dual " " " = 2.08e-10
rel. primal infeas (unscaled problem) = 0.00e+00
rel. dual " " " = 0.00e+00
norm(X), norm(y), norm(Z) = 2.2e+00, 5.8e-01, 2.8e+00
norm(A), norm(b), norm(C) = 5.5e+00, 2.0e+00, 3.8e+00
Total CPU time (secs) = 0.13
CPU time per iteration = 0.01
termination code = 0
DIMACS: 2.1e-11 0.0e+00 4.0e-10 0.0e+00 1.2e-08 1.3e-08
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -0.39282
Done!
--------------------------------------------------------------------------------
The minimum and maximum values of rho_14 are: -0.39282 and 0.22991
with corresponding correlation matrices:
1.0000 0.6000 0.8433 -0.3928
0.6000 1.0000 0.3322 0.5000
0.8433 0.3322 1.0000 -0.5311
-0.3928 0.5000 -0.5311 1.0000
1.0000 0.7127 0.8000 0.2299
0.7127 1.0000 0.3120 0.5827
0.8000 0.3120 1.0000 -0.4000
0.2299 0.5827 -0.4000 1.0000