% 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