Separating ellipsoids in 2D
n = 2;
A = eye(n);
b = zeros(n,1);
C = [2 1; -.5 1];
d = [-3; -3];
cvx_begin
variables x(n) y(n) w(n)
dual variables lam muu z
minimize ( norm(w,2) )
subject to
lam: square_pos( norm (A*x + b) ) <= 1;
muu: square_pos( norm (C*y + d) ) <= 1;
z: x - y == w;
cvx_end
t = (x + y)/2;
p=z;
p(1) = z(2); p(2) = -z(1);
c = linspace(-2,2,100);
q = repmat(t,1,length(c)) +p*c;
nopts = 1000;
angles = linspace(0,2*pi,nopts);
[u,v] = meshgrid([-2:0.01:4]);
z1 = (A(1,1)*u + A(1,2)*v + b(1)).^2 + (A(2,1)*u + A(2,2)*v + b(2)).^2;
z2 = (C(1,1)*u + C(1,2)*v + d(1)).^2 + (C(2,1)*u + C(2,2)*v + d(2)).^2;
contour(u,v,z1,[1 1]);
hold on;
contour(u,v,z2,[1 1]);
axis square
plot(x(1),x(2),'r+');
plot(y(1),y(2),'b+');
line([x(1) y(1)],[x(2) y(2)]);
plot(q(1,:),q(2,:),'k');
Calling sedumi: 17 variables, 9 equality constraints
For improved efficiency, sedumi is solving the dual problem.
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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 = 9, order n = 13, dim = 20, blocks = 6
nnz(A) = 19 + 0, nnz(ADA) = 55, nnz(L) = 32
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 4.34E+00 0.000
1 : -1.99E-01 1.38E+00 0.000 0.3184 0.9000 0.9000 2.23 1 1 1.6E+00
2 : -1.09E+00 1.26E-01 0.000 0.0914 0.9900 0.9900 1.22 1 1 1.5E-01
3 : -1.19E+00 3.13E-03 0.000 0.0247 0.9900 0.9900 1.11 1 1 3.0E-03
4 : -1.19E+00 8.33E-06 0.249 0.0027 0.9990 0.9990 1.00 1 1 8.0E-06
5 : -1.19E+00 5.75E-07 0.000 0.0691 0.9900 0.9900 1.00 1 1 5.6E-07
6 : -1.19E+00 8.33E-08 0.000 0.1448 0.9105 0.9000 1.00 1 1 8.4E-08
7 : -1.19E+00 3.15E-09 0.375 0.0378 0.9902 0.9900 1.00 1 1 3.2E-09
iter seconds digits c*x b*y
7 0.0 8.9 -1.1924413509e+00 -1.1924413525e+00
|Ax-b| = 9.6e-10, [Ay-c]_+ = 3.2E-09, |x|= 2.9e+00, |y|= 3.0e+00
Detailed timing (sec)
Pre IPM Post
1.000E-02 3.000E-02 0.000E+00
Max-norms: ||b||=1, ||c|| = 3.750000e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 14.478.
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Status: Solved
Optimal value (cvx_optval): +1.19244