% Exercise 4.47: Maximum determinant PSD matrix completion % Boyd & Vandenberghe "Convex Optimization" % Almir Mutapcic - Jan 2006 % % Given a symmetric matrix A in R^(n-by-n) with some entries unspecified % we find its completion such that A is positive semidefinite and % it has a maximum determinant out of all possible completions. % This problem can be formulated as a log det (and det_rootn) problem. % % This is a numerical instance of the specified book exercise. % problem size n = 4; % create and solve the problem cvx_begin sdp % A is a PSD symmetric matrix (n-by-n) variable A(n,n) symmetric; A >= 0; % constrained matrix entries. A(1,1) == 3; A(2,2) == 2; A(3,3) == 1; A(4,4) == 5; % Note that because A is symmetric, these off-diagonal % constraints affect the corresponding element on the % opposite side of the diagonal. A(1,2) == .5; A(1,4) == .25; A(2,3) == .75; % find the solution to the problem maximize( log_det( A ) ) % maximize( det_rootn( A ) ) cvx_end % display solution disp(['Matrix A with maximum determinant (' num2str(det(A)) ') is:']) A disp(['Its eigenvalues are:']) eigs = eig(A)