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 sedumi: 18 variables, 6 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 6, order n = 13, dim = 25, blocks = 2
nnz(A) = 14 + 0, nnz(ADA) = 36, nnz(L) = 21
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 6.09E+00 0.000
1 : 4.91E-01 1.80E+00 0.000 0.2954 0.9000 0.9000 1.63 1 1 1.6E+00
2 : 3.94E-01 4.91E-01 0.000 0.2726 0.9000 0.9000 1.48 1 1 5.5E-01
3 : 2.46E-01 3.95E-02 0.000 0.0806 0.9900 0.9900 1.55 1 1 7.0E-02
4 : 2.31E-01 2.31E-03 0.355 0.0584 0.9900 0.9900 1.08 1 1 4.5E-03
5 : 2.30E-01 9.41E-05 0.000 0.0408 0.9900 0.9900 1.00 1 1 1.8E-04
6 : 2.30E-01 3.88E-07 0.292 0.0041 0.9990 0.9459 1.00 1 1 1.2E-06
7 : 2.30E-01 8.10E-09 0.000 0.0209 0.9900 0.9900 1.00 3 3 2.5E-08
8 : 2.30E-01 1.59E-09 0.113 0.1968 0.9000 0.9000 1.00 3 3 4.9E-09
iter seconds digits c*x b*y
8 0.0 Inf 2.2990908942e-01 2.2990909282e-01
|Ax-b| = 2.9e-10, [Ay-c]_+ = 3.1E-09, |x|= 2.8e+00, |y|= 1.4e+00
Detailed timing (sec)
Pre IPM Post
0.000E+00 4.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=1, |skip| = 0, ||L.L|| = 3230.45.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.229909
Done!
Solving the lower bound SDP ...
Calling sedumi: 18 variables, 6 equality constraints
For improved efficiency, sedumi is solving the dual problem.
------------------------------------------------------------
SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 6, order n = 13, dim = 25, blocks = 2
nnz(A) = 14 + 0, nnz(ADA) = 36, nnz(L) = 21
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 6.09E+00 0.000
1 : 5.01E-01 1.85E+00 0.000 0.3033 0.9000 0.9000 1.63 1 1 1.6E+00
2 : 4.85E-01 5.25E-01 0.000 0.2843 0.9000 0.9000 1.52 1 1 5.3E-01
3 : 4.03E-01 4.95E-02 0.000 0.0943 0.9900 0.9900 1.62 1 1 5.5E-02
4 : 3.93E-01 9.96E-04 0.000 0.0201 0.9900 0.9900 1.11 1 1 1.1E-03
5 : 3.93E-01 5.17E-05 0.145 0.0520 0.9900 0.9900 1.00 1 1 5.7E-05
6 : 3.93E-01 7.15E-07 0.000 0.0138 0.9900 0.8308 1.00 1 1 1.8E-06
7 : 3.93E-01 2.36E-08 0.224 0.0331 0.9900 0.9900 1.00 1 1 5.9E-08
8 : 3.93E-01 4.65E-09 0.000 0.1969 0.9000 0.9000 1.00 1 1 1.2E-08
iter seconds digits c*x b*y
8 0.0 Inf 3.9282033240e-01 3.9282034134e-01
|Ax-b| = 5.2e-10, [Ay-c]_+ = 7.0E-09, |x|= 2.2e+00, |y|= 1.4e+00
Detailed timing (sec)
Pre IPM Post
0.000E+00 4.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 836.636.
------------------------------------------------------------
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 0.8482 -0.3928
0.6000 1.0000 0.2946 0.5000
0.8482 0.2946 1.0000 -0.5795
-0.3928 0.5000 -0.5795 1.0000
1.0000 0.7270 0.8000 0.2299
0.7270 1.0000 0.3182 0.5944
0.8000 0.3182 1.0000 -0.4000
0.2299 0.5944 -0.4000 1.0000