# 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);

    Successive approximation method to be employed.
       For improved efficiency, SDPT3 is solving the dual problem.
       SDPT3 will be called several times to refine the solution.
       Original size: 155 variables, 60 equality constraints
       50 exponentials add 400 variables, 250 equality constraints
    -----------------------------------------------------------------
     Cones  |             Errors              |
    Mov/Act | Centering  Exp cone   Poly cone | Status
    --------+---------------------------------+---------
     50/ 50 | 3.700e+00  7.922e-01  0.000e+00 | Solved
     50/ 50 | 6.027e-01  2.364e-02  0.000e+00 | Solved
     46/ 50 | 5.799e-02  2.176e-04  0.000e+00 | Solved
     13/ 39 | 6.785e-03  2.986e-06  0.000e+00 | Solved
      2/ 12 | 7.432e-04  3.537e-08  0.000e+00 | Solved
      0/  1 | 8.438e-05  6.251e-11  0.000e+00 | 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
        0.4031
        0.1248