# Exercise 4.47: Maximum determinant PSD matrix completion

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    % 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)

    Successive approximation method to be employed.
       sedumi will be called several times to refine the solution.
       Original size: 59 variables, 41 equality constraints
       1 exponentials add 8 variables, 5 equality constraints
    -----------------------------------------------------------------
     Cones  |             Errors              |
    Mov/Act | Centering  Exp cone   Poly cone | Status
    --------+---------------------------------+---------
      1/  1 | 7.261e-01  3.688e-02  6.009e-08 | Solved
      1/  1 | 2.810e-02  5.643e-05  2.084e-07 | Solved
      1/  1 | 1.731e-03  4.515e-07  2.323e-07 | Solved
      0/  1 | 1.081e-04  2.340e-07  2.327e-07 | Solved
    -----------------------------------------------------------------
    Status: Solved
    Optimal value (cvx_optval): +3.02422
    Matrix A with maximum determinant (20.578) is:

    A =

        3.0000    0.5000    0.1875    0.2500
        0.5000    2.0000    0.7500    0.0417
        0.1875    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