Minimum volume ellipsoid covering union of ellipsoids
As = {}; bs = {}; cs = {};
As{1} = [ 0.1355 0.1148; 0.1148 0.4398];
As{2} = [ 0.6064 -0.1022; -0.1022 0.7344];
As{3} = [ 0.7127 -0.0559; -0.0559 0.9253];
As{4} = [ 0.2706 -0.1379; -0.1379 0.2515];
As{5} = [ 0.4008 -0.1112; -0.1112 0.2107];
bs{1} = [ -0.2042 0.0264]';
bs{2} = [ 0.8259 -2.1188]';
bs{3} = [ -0.0256 1.0591]';
bs{4} = [ 0.1827 -0.3844]';
bs{5} = [ 0.3823 -0.8253]';
cs{1} = 0.2351;
cs{2} = 5.8250;
cs{3} = 0.9968;
cs{4} = -0.2981;
cs{5} = 2.6735;
n = 2;
m = size(bs,2);
cvx_begin sdp
variable Asqr(n,n) symmetric
variable btilde(n)
variable t(m)
maximize( det_rootn( Asqr ) )
subject to
t >= 0;
for i = 1:m
[ -(Asqr - t(i)*As{i}), -(btilde - t(i)*bs{i}), zeros(n,n);
-(btilde - t(i)*bs{i})', -(- 1 - t(i)*cs{i}), -btilde';
zeros(n,n), -btilde, Asqr] >= 0;
end
cvx_end
A = sqrtm(Asqr);
b = A\btilde;
noangles = 200;
angles = linspace( 0, 2 * pi, noangles );
clf
for i=1:m
Ai = sqrtm(As{i}); bi = Ai\bs{i};
alpha = bs{i}'*inv(As{i})*bs{i} - cs{i};
ellipse = Ai \ [ sqrt(alpha)*cos(angles)-bi(1) ; sqrt(alpha)*sin(angles)-bi(2) ];
plot( ellipse(1,:), ellipse(2,:), 'b-' );
hold on
end
ellipse = A \ [ cos(angles) - b(1) ; sin(angles) - b(2) ];
plot( ellipse(1,:), ellipse(2,:), 'r--' );
axis square
axis off
hold off
Calling SeDuMi: 93 variables (0 free), 79 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
eqs m = 79, order n = 37, dim = 151, blocks = 8
nnz(A) = 161 + 0, nnz(ADA) = 2793, nnz(L) = 1580
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 1.87E+000 0.000
1 : -6.85E-001 5.30E-001 0.000 0.2830 0.9000 0.9000 1.03 1 1 3.7E+000
2 : -3.45E-001 1.70E-001 0.000 0.3214 0.9000 0.9000 1.82 1 1 9.0E-001
3 : -1.20E-001 4.40E-002 0.000 0.2583 0.9000 0.9000 1.89 1 1 1.9E-001
4 : -8.02E-002 8.07E-003 0.000 0.1833 0.9000 0.9000 1.35 1 1 3.3E-002
5 : -7.96E-002 1.86E-003 0.000 0.2304 0.9000 0.9000 0.95 1 1 7.8E-003
6 : -7.87E-002 5.35E-005 0.000 0.0288 0.9902 0.9900 0.99 1 1 3.2E-004
7 : -7.87E-002 1.39E-006 0.000 0.0261 0.9900 0.9900 1.00 1 1 7.3E-006
8 : -7.87E-002 1.21E-007 0.499 0.0866 0.9900 0.9900 1.00 1 1 6.5E-007
9 : -7.87E-002 2.42E-008 0.000 0.2005 0.9000 0.8355 1.00 1 1 1.3E-007
10 : -7.87E-002 1.57E-009 0.414 0.0647 0.9900 0.9901 1.00 2 2 7.7E-009
iter seconds digits c*x b*y
10 0.1 8.0 -7.8681465304e-002 -7.8681466096e-002
|Ax-b| = 5.1e-009, [Ay-c]_+ = 6.8E-010, |x|= 6.3e+000, |y|= 2.2e+000
Detailed timing (sec)
Pre IPM Post
4.006E-002 7.010E-002 1.001E-002
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 2030.47.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.0786815