a1 = [ 2; 1];
a2 = [ 2; -1];
a3 = [-1; 2];
a4 = [-1; -2];
b = ones(4,1);
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
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.
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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 = 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.
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Status: Solved
Optimal value (cvx_optval): +0.447214