# Section 8.5.3: Analytic center of a set of linear inequalities

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    % Boyd & Vandenberghe "Convex Optimization"
    % JoÃ«lle Skaf - 04/29/08
    %
    % The analytic center of a set of linear inequalities and equalities:
    %           a_i^Tx <= b_i   i=1,...,m,
    %           Fx = g,
    % is the solution of the unconstrained minimization problem
    %           minimize    -sum_{i=1}^m log(b_i-a_i^Tx).

    % Input data
    randn('state', 0);
    rand('state', 0);
    n = 10;
    m = 50;
    p = 5;
    tmp = randn(n,1);
    A = randn(m,n);
    b = A*tmp + 10*rand(m,1);
    F = randn(p,n);
    g = F*tmp;

    % Analytic center
    cvx_begin
        variable x(n)
        minimize -sum(log(b-A*x))
        F*x == g
    cvx_end

    disp(['The analytic center of the set of linear inequalities and ' ...
          'equalities is: ']);
    disp(x);

    CVX Warning:
       Models involving "log" or other functions in the log, exp, and entropy
       family are solved using an experimental successive approximation method.
       The method requires multiple calls to the solver, so it can be slow; and
       in certain cases it fails to converge. See Appendix D of the the user's
       guide for more information about this method, and for instructions on how
       to suppress this warning in the future.

    Successive approximation method to be employed.
       sedumi will be called several times to refine the solution.
       Original size: 160 variables, 105 equality constraints
       50 exponentials add 400 variables, 250 equality constraints
    -----------------------------------------------------------------
     Cones  |             Errors              |
    Mov/Act | Centering  Exp cone   Poly cone | Status
    --------+---------------------------------+---------
     50/ 50 | 2.303e+00  3.573e-01  5.090e-08 | Solved
     49/ 50 | 1.788e-01  2.265e-03  2.233e-07 | Solved
     30/ 50 | 1.486e-02  1.622e-05  2.066e-07 | Solved
      0/ 50 | 8.671e-04  2.138e-07  2.038e-07 | Solved
    -----------------------------------------------------------------
    Status: Solved
    Optimal value (cvx_optval): -64.8504
    The analytic center of the set of linear inequalities and equalities is:
       -0.3618
       -1.5333
        0.1387
        0.2491
       -1.1163
        1.3142
        1.2303
       -0.0511
        0.4030
        0.1248