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 (0 free), 12 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 12, order n = 13, dim = 25, blocks = 2
nnz(A) = 20 + 0, nnz(ADA) = 144, nnz(L) = 78
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.49E+000 0.000
1 : -2.96E-001 4.40E-001 0.000 0.2954 0.9000 0.9000 1.63 1 1 1.2E+000
2 : -4.38E-001 1.20E-001 0.000 0.2726 0.9000 0.9000 1.48 1 1 3.0E-001
3 : -2.52E-001 9.66E-003 0.000 0.0806 0.9900 0.9900 1.55 1 1 2.4E-002
4 : -2.31E-001 5.64E-004 0.355 0.0584 0.9900 0.9900 1.08 1 1 1.4E-003
5 : -2.30E-001 2.30E-005 0.000 0.0408 0.9900 0.9900 1.00 1 1 5.7E-005
6 : -2.30E-001 1.31E-007 0.325 0.0057 0.9853 0.9990 1.00 1 1 5.0E-007
7 : -2.30E-001 8.11E-009 0.000 0.0617 0.9900 0.9900 1.00 1 1 3.1E-008
8 : -2.30E-001 2.91E-010 0.389 0.0358 0.9900 0.9900 1.00 2 2 1.1E-009
iter seconds digits c*x b*y
8 0.1 Inf -2.2990908487e-001 -2.2990908453e-001
|Ax-b| = 1.7e-009, [Ay-c]_+ = 2.4E-010, |x|= 2.8e+000, |y|= 2.3e+000
Detailed timing (sec)
Pre IPM Post
1.001E-002 9.013E-002 0.000E+000
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1159.47.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.229909
Done!
Solving the lower bound SDP ...
Calling SeDuMi: 18 variables (0 free), 12 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 12, order n = 13, dim = 25, blocks = 2
nnz(A) = 20 + 0, nnz(ADA) = 144, nnz(L) = 78
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.49E+000 0.000
1 : -3.13E-001 4.52E-001 0.000 0.3033 0.9000 0.9000 1.63 1 1 1.2E+000
2 : -5.41E-001 1.28E-001 0.000 0.2843 0.9000 0.9000 1.52 1 1 3.1E-001
3 : -4.09E-001 1.21E-002 0.000 0.0943 0.9900 0.9900 1.62 1 1 2.3E-002
4 : -3.93E-001 2.43E-004 0.000 0.0201 0.9900 0.9900 1.11 1 1 4.4E-004
5 : -3.93E-001 1.26E-005 0.145 0.0520 0.9900 0.9900 1.00 1 1 2.3E-005
6 : -3.93E-001 1.76E-007 0.000 0.0139 0.9574 0.9900 1.00 1 1 5.8E-007
7 : -3.93E-001 6.56E-009 0.030 0.0373 0.9900 0.9815 1.00 1 1 2.2E-008
8 : -3.93E-001 1.29E-009 0.061 0.1970 0.8975 0.9000 1.00 2 2 4.3E-009
iter seconds digits c*x b*y
8 0.0 Inf -3.9282032728e-001 -3.9282032607e-001
|Ax-b| = 6.6e-009, [Ay-c]_+ = 9.6E-010, |x|= 2.8e+000, |y|= 1.8e+000
Detailed timing (sec)
Pre IPM Post
0.000E+000 2.003E-002 0.000E+000
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 2421.04.
------------------------------------------------------------------------
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 0.8486 -0.3928
0.6000 1.0000 0.2940 0.5000
0.8486 0.2940 1.0000 -0.5807
-0.3928 0.5000 -0.5807 1.0000
1.0000 0.7291 0.8000 0.2299
0.7291 1.0000 0.3202 0.5943
0.8000 0.3202 1.0000 -0.4000
0.2299 0.5943 -0.4000 1.0000