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R1252/simulation-code/cvx/examples/cvxbook/Ch08_geometric_probs/ex_8_3.m
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% Example 8.3: Bounding correlation coefficients
% Boyd & Vandenberghe "Convex Optimization"
% Joelle Skaf - 10/09/05
%
% Let C be a correlation matrix. Given lower and upper bounds on
% some of the angles (or correlation coeff.), find the maximum and minimum
% possible values of rho_14 by solving 2 SDP's
% minimize/maximize rho_14
% s.t. C >=0
% 0.6 <= rho_12 <= 0.9
% 0.8 <= rho_13 <= 0.9
% 0.5 <= rho_24 <= 0.7
% -0.8 <= rho_34 <= -0.4
n = 4;
% Upper bound SDP
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');
% Lower bound SDP
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');
% Displaying results
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)
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