Example 8.3: Bounding correlation coefficients
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: 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 mean(obj) cputime
-------------------------------------------------------------------
0|0.000|0.000|5.0e-01|2.4e+00|1.8e+02| 9.455844e+00| 0:0:00| chol 1 1
1|1.000|0.965|1.2e-06|1.7e-01|1.6e+01| 3.569251e+00| 0:0:00| chol 1 1
2|0.984|1.000|1.7e-06|9.0e-03|2.0e+00| 1.040621e+00| 0:0:00| chol 1 1
3|0.928|1.000|1.9e-07|9.0e-04|1.5e-01| 2.644014e-01| 0:0:00| chol 1 1
4|0.951|0.987|1.2e-07|1.0e-04|8.2e-03| 2.328182e-01| 0:0:00| chol 1 1
5|0.956|0.992|1.1e-08|9.7e-06|4.4e-04| 2.300940e-01| 0:0:00| chol 1 1
6|0.958|0.991|3.7e-09|9.8e-07|2.0e-05| 2.299200e-01| 0:0:00| chol 1 1
7|0.997|1.000|1.8e-09|7.5e-10|1.0e-06| 2.299093e-01| 0:0:00| chol 1 1
8|1.000|1.000|1.6e-09|3.6e-10|8.3e-08| 2.299091e-01| 0:0:00| chol 1 1
9|1.000|1.000|2.7e-10|7.0e-11|4.0e-09| 2.299091e-01| 0:0:00|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 9
primal objective value = 2.29909086e-01
dual objective value = 2.29909082e-01
gap := trace(XZ) = 3.95e-09
relative gap = 2.71e-09
actual relative gap = 2.12e-09
rel. primal infeas = 2.73e-10
rel. dual infeas = 6.96e-11
norm(X), norm(y), norm(Z) = 2.8e+00, 1.3e+00, 2.8e+00
norm(A), norm(b), norm(C) = 5.5e+00, 2.0e+00, 3.9e+00
Total CPU time (secs) = 0.2
CPU time per iteration = 0.0
termination code = 0
DIMACS: 2.7e-10 0.0e+00 1.3e-10 0.0e+00 2.1e-09 2.7e-09
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.229909
Done!
Solving the lower bound SDP ...
Calling SDPT3: 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 mean(obj) cputime
-------------------------------------------------------------------
0|0.000|0.000|5.0e-01|2.4e+00|1.8e+02| 9.455844e+00| 0:0:00| chol 1 1
1|1.000|0.978|1.2e-06|1.4e-01|1.4e+01| 3.594871e+00| 0:0:00| chol 1 1
2|0.990|1.000|1.4e-06|9.0e-03|1.7e+00| 1.044015e+00| 0:0:00| chol 1 1
3|0.912|1.000|1.5e-07|9.0e-04|1.5e-01| 4.325185e-01| 0:0:00| chol 1 1
4|0.993|0.994|5.5e-08|9.4e-05|5.5e-03| 3.943689e-01| 0:0:00| chol 1 1
5|0.934|0.985|5.3e-09|1.0e-05|3.1e-04| 3.929550e-01| 0:0:00| chol 1 1
6|1.000|1.000|1.7e-08|9.0e-07|3.8e-05| 3.928326e-01| 0:0:00| chol 1 1
7|0.970|0.983|2.8e-09|1.6e-08|1.1e-06| 3.928207e-01| 0:0:00| chol 1 1
8|1.000|1.000|3.4e-09|5.6e-10|1.7e-07| 3.928204e-01| 0:0:00| chol 1 1
9|1.000|1.000|2.8e-10|1.1e-10|5.3e-09| 3.928203e-01| 0:0:00|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 9
primal objective value = 3.92820326e-01
dual objective value = 3.92820322e-01
gap := trace(XZ) = 5.28e-09
relative gap = 2.96e-09
actual relative gap = 2.37e-09
rel. primal infeas = 2.78e-10
rel. dual infeas = 1.13e-10
norm(X), norm(y), norm(Z) = 2.2e+00, 1.4e+00, 2.8e+00
norm(A), norm(b), norm(C) = 5.5e+00, 2.0e+00, 3.9e+00
Total CPU time (secs) = 0.2
CPU time per iteration = 0.0
termination code = 0
DIMACS: 2.8e-10 0.0e+00 2.2e-10 0.0e+00 2.4e-09 3.0e-09
-------------------------------------------------------------------
------------------------------------------------------------
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 0.8239 -0.3928
0.6000 1.0000 0.2979 0.5000
0.8239 0.2979 1.0000 -0.5494
-0.3928 0.5000 -0.5494 1.0000
1.0000 0.6907 0.8000 0.2299
0.6907 1.0000 0.2994 0.5694
0.8000 0.2994 1.0000 -0.4000
0.2299 0.5694 -0.4000 1.0000