# Section 4.3.1: Compute and display the Chebyshev center of a 2D polyhedron

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    % Boyd & Vandenberghe, "Convex Optimization"
    % Joëlle Skaf - 08/16/05
    % (a figure is generated)
    %
    % The goal is to find the largest Euclidean ball (i.e. its center and
    % radius) that lies in a polyhedron described by linear inequalites in this
    % fashion: P = {x : a_i'*x <= b_i, i=1,...,m} where x is in R^2

    % Generate the input data
    a1 = [ 2;  1];
    a2 = [ 2; -1];
    a3 = [-1;  2];
    a4 = [-1; -2];
    b = ones(4,1);

    % Create and solve the model
    cvx_begin
        variable r(1)
        variable x_c(2)
        maximize ( r )
        a1'*x_c + r*norm(a1,2) <= b(1);
        a2'*x_c + r*norm(a2,2) <= b(2);
        a3'*x_c + r*norm(a3,2) <= b(3);
        a4'*x_c + r*norm(a4,2) <= b(4);
    cvx_end

    % Generate the figure
    x = linspace(-2,2);
    theta = 0:pi/100:2*pi;
    plot( x, -x*a1(1)./a1(2) + b(1)./a1(2),'b-');
    hold on
    plot( x, -x*a2(1)./a2(2) + b(2)./a2(2),'b-');
    plot( x, -x*a3(1)./a3(2) + b(3)./a3(2),'b-');
    plot( x, -x*a4(1)./a4(2) + b(4)./a4(2),'b-');
    plot( x_c(1) + r*cos(theta), x_c(2) + r*sin(theta), 'r');
    plot(x_c(1),x_c(2),'k+')
    xlabel('x_1')
    ylabel('x_2')
    title('Largest Euclidean ball lying in a 2D polyhedron');
    axis([-1 1 -1 1])
    axis equal

    Calling sedumi: 4 variables, 3 equality constraints
       For improved efficiency, sedumi is solving the dual problem.
    ------------------------------------------------------------
    SeDuMi 1.21 by AdvOL, 2005-2008 and Jos F. Sturm, 1998-2003.
    Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
    eqs m = 3, order n = 5, dim = 5, blocks = 1
    nnz(A) = 12 + 0, nnz(ADA) = 9, nnz(L) = 6
     it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
      0 :            4.47E+01 0.000
      1 :  -6.25E-02 1.08E+01 0.000 0.2426 0.9000 0.9000   1.41  1  1  4.0E+00
      2 :   4.05E-01 2.36E+00 0.000 0.2180 0.9000 0.9000   2.92  1  1  3.7E-01
      3 :   4.46E-01 6.75E-02 0.000 0.0286 0.9900 0.9900   1.38  1  1  8.6E-03
      4 :   4.47E-01 2.06E-06 0.070 0.0000 1.0000 1.0000   1.01  1  1
    iter seconds digits       c*x               b*y
      4      0.0   Inf  4.4721359550e-01  4.4721359550e-01
    |Ax-b| =   5.4e-17, [Ay-c]_+ =   2.2E-16, |x|=  2.4e-01, |y|=  4.5e-01

    Detailed timing (sec)
       Pre          IPM          Post
    0.000E+00    3.000E-02    0.000E+00
    Max-norms: ||b||=1, ||c|| = 1,
    Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.
    ------------------------------------------------------------
    Status: Solved
    Optimal value (cvx_optval): +0.447214