Exercise 4.47: Maximum determinant PSD matrix completion
n = 4;
cvx_begin sdp
variable A(n,n) symmetric;
A >= 0;
A(1,1) == 3;
A(2,2) == 2;
A(3,3) == 1;
A(4,4) == 5;
A(1,2) == .5;
A(1,4) == .25;
A(2,3) == .75;
maximize( det_rootn( A ) )
cvx_end
disp(['Matrix A with maximum determinant (' num2str(det(A)) ') is:'])
A
disp(['Its eigenvalues are:'])
eigs = eig(A)
Calling SeDuMi: 55 variables (0 free), 39 equality constraints
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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 = 39, order n = 19, dim = 93, blocks = 6
nnz(A) = 59 + 0, nnz(ADA) = 1109, nnz(L) = 578
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.12E+000 0.000
1 : -1.69E+000 2.85E-001 0.000 0.2534 0.9000 0.9000 1.75 1 1 1.0E+000
2 : -1.31E+000 7.50E-002 0.000 0.2637 0.9000 0.9000 1.66 1 1 1.9E-001
3 : -1.95E+000 1.55E-002 0.000 0.2066 0.9000 0.9000 0.88 1 1 3.6E-002
4 : -2.13E+000 5.60E-004 0.000 0.0361 0.9900 0.9900 0.94 1 1 1.3E-003
5 : -2.13E+000 1.69E-005 0.000 0.0301 0.9900 0.9900 1.00 1 1 4.0E-005
6 : -2.13E+000 4.66E-007 0.000 0.0276 0.9900 0.9480 1.00 1 1 1.5E-006
7 : -2.13E+000 9.81E-009 0.104 0.0211 0.9900 0.9900 1.00 1 1 3.3E-008
8 : -2.13E+000 1.50E-010 0.000 0.0153 0.9906 0.9900 1.00 2 2 1.6E-009
iter seconds digits c*x b*y
8 0.0 Inf -2.1298576989e+000 -2.1298576958e+000
|Ax-b| = 9.4e-010, [Ay-c]_+ = 1.2E-009, |x|= 1.7e+001, |y|= 2.5e+000
Detailed timing (sec)
Pre IPM Post
2.003E-002 4.006E-002 0.000E+000
Max-norms: ||b||=5, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 788.835.
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Status: Solved
Optimal value (cvx_optval): +2.12986
Matrix A with maximum determinant (20.578) is:
A =
3.0000 0.5000 0.1874 0.2500
0.5000 2.0000 0.7500 0.0417
0.1874 0.7500 1.0000 0.0156
0.2500 0.0417 0.0156 5.0000
Its eigenvalues are:
eigs =
0.5964
2.0908
3.2773
5.0355