# Weighted analytic center of a set of linear inequalities

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    % JoÃ«lle Skaf - 04/29/08
    %
    % The weighted analytic center of a set of linear inequalities:
    %           a_i^Tx <= b_i   i=1,...,m,
    % is the solution of the unconstrained minimization problem
    %           minimize    -sum_{i=1}^m w_i*log(b_i-a_i^Tx),
    % where w_i>0

    % Input data
    randn('state', 0);
    rand('state', 0);
    n = 10;
    m = 50;
    tmp = randn(n,1);
    A = randn(m,n);
    b = A*tmp + 2*rand(m,1);
    w = rand(m,1);

    % Analytic center
    cvx_begin
        variable x(n)
        minimize -sum(w.*log(b-A*x))
    cvx_end

    disp('The weighted analytic center of the set of linear inequalities 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: 150 variables, 60 equality constraints
       50 exponentials add 400 variables, 250 equality constraints
    -----------------------------------------------------------------
     Cones  |             Errors              |
    Mov/Act | Centering  Exp cone   Poly cone | Status
    --------+---------------------------------+---------
     50/ 50 | 4.642e+00  1.215e+00  0.000e+00 | Solved
     50/ 50 | 9.246e-01  5.647e-02  0.000e+00 | Solved
     47/ 50 | 7.922e-02  4.134e-04  0.000e+00 | Solved
     35/ 44 | 8.288e-03  4.540e-06  0.000e+00 | Solved
     11/ 30 | 9.917e-04  6.459e-08  0.000e+00 | Solved
      0/  5 | 1.158e-04  4.969e-10  0.000e+00 | Solved
    -----------------------------------------------------------------
    Status: Solved
    Optimal value (cvx_optval): +5.99254

    The weighted analytic center of the set of linear inequalities is:
       -0.5100
       -1.4794
        0.3397
        0.1944
       -1.0444
        1.1956
        1.3927
       -0.2815
        0.2862
        0.3779