Euclidean distance between polyhedra in 2D
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...');
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');
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);
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: 11 variables, 6 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 = 6, order n = 11, dim = 12, blocks = 6
nnz(A) = 11 + 0, nnz(ADA) = 24, nnz(L) = 15
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 6.13E+00 0.000
1 : -5.77E-01 2.08E+00 0.000 0.3392 0.9000 0.9000 2.26 1 1 1.8E+00
2 : -1.83E+00 5.31E-01 0.000 0.2555 0.9000 0.9000 1.21 1 1 3.8E-01
3 : -2.11E+00 2.87E-02 0.000 0.0539 0.9900 0.9900 1.21 1 1 1.7E-02
4 : -2.12E+00 5.78E-03 0.000 0.2017 0.9000 0.9000 1.02 1 1 3.4E-03
5 : -2.12E+00 1.77E-04 0.000 0.0306 0.9900 0.9900 1.00 1 1 1.3E-04
6 : -2.12E+00 1.18E-05 0.446 0.0664 0.9900 0.9900 1.00 1 1 8.3E-06
7 : -2.12E+00 8.56E-08 0.000 0.0073 0.9902 0.9900 1.00 1 1 1.2E-07
8 : -2.12E+00 1.02E-10 0.016 0.0012 0.9990 0.9874 1.00 1 1 1.0E-09
iter seconds digits c*x b*y
8 0.0 Inf -2.1213203424e+00 -2.1213203412e+00
|Ax-b| = 6.2e-11, [Ay-c]_+ = 1.2E-09, |x|= 2.4e+00, |y|= 3.6e+00
Detailed timing (sec)
Pre IPM Post
0.000E+00 4.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 4,
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