% Section 8.2.1, Boyd & Vandenberghe "Convex Optimization"
% Joelle Skaf - 10/09/05
% (a figure is generated)
%
% Given two polyhedra C = {x | A1*x <= b1} and D = {x | A2*x <= b2}, the
% distance between them is the optimal value of the problem:
%           minimize    || x - y ||_2
%               s.t.    A1*x <= b1
%                       A2*y <= b2
% Note: here x is in R^2

% Input data
randn('seed',0);
n = 2;
m = 2*n;
A1 = randn(m,n);
b1 = randn(m,1);
A2 = randn(m,n);
b2 = randn(m,1);

fprintf(1,'Computing the distance between the 2 polyhedra...');
% Solution via CVX
cvx_begin
    variables x(n) y(n)
    minimize (norm(x - y))
    norm(x,1) <= 2;
    norm(y-[4;3],inf) <=1;
cvx_end

fprintf(1,'Done! \n');

% Displaying results
disp('------------------------------------------------------------------');
disp('The distance between the 2 polyhedra C and D is: ' );
disp(['dist(C,D) = ' num2str(cvx_optval)]);
disp('The optimal points are: ')
disp('x = '); disp(x);
disp('y = '); disp(y);

%Plotting
figure;
fill([-2; 0; 2; 0],[0;2;0;-2],'b', [3;5;5;3],[2;2;4;4],'r')
axis([-3 6 -3 6])
axis square
hold on;
plot(x(1),x(2),'k.')
plot(y(1),y(2),'k.')
plot([x(1) y(1)],[x(2) y(2)])
title('Euclidean distance between 2 polyhedron in R^2');
xlabel('x_1');
ylabel('x_2');
Computing the distance between the 2 polyhedra... 
Calling sedumi: 15 variables, 5 equality constraints
------------------------------------------------------------
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 = 5, order n = 15, dim = 16, blocks = 2
nnz(A) = 23 + 0, nnz(ADA) = 17, nnz(L) = 12
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            5.33E+00 0.000
  1 :  -1.07E-01 1.65E+00 0.000 0.3105 0.9000 0.9000   2.24  1  1  2.6E+00
  2 :   1.35E+00 4.81E-01 0.000 0.2907 0.9000 0.9000   0.96  1  1  7.8E-01
  3 :   2.01E+00 1.02E-01 0.000 0.2131 0.9000 0.9000   1.34  1  1  1.4E-01
  4 :   2.12E+00 3.39E-03 0.000 0.0331 0.9900 0.9900   1.05  1  1  4.5E-03
  5 :   2.12E+00 2.90E-04 0.414 0.0855 0.9900 0.9900   1.00  1  1  3.8E-04
  6 :   2.12E+00 5.11E-06 0.000 0.0176 0.9900 0.9903   1.00  1  1  1.3E-05
  7 :   2.12E+00 3.04E-08 0.327 0.0060 0.9966 0.9990   1.00  1  1  5.9E-08
  8 :   2.12E+00 9.48E-10 0.000 0.0312 0.9900 0.9902   1.00  1  1  2.2E-09

iter seconds digits       c*x               b*y
  8      0.1   9.8  2.1213203395e+00  2.1213203392e+00
|Ax-b| =   8.0e-09, [Ay-c]_+ =   0.0E+00, |x|=  3.7e+00, |y|=  1.6e+00

Detailed timing (sec)
   Pre          IPM          Post
1.000E-02    1.000E-01    0.000E+00    
Max-norms: ||b||=4, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +2.12132
Done! 
------------------------------------------------------------------
The distance between the 2 polyhedra C and D is: 
dist(C,D) = 2.1213
The optimal points are: 
x = 
    1.5000
    0.5000

y = 
    3.0000
    2.0000