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 SDPT3: 93 variables, 14 equality constraints
For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------
num. of constraints = 14
dim. of sdp var = 31, num. of sdp blk = 7
dim. of linear var = 5
*******************************************************************
SDPT3: Infeasible path-following algorithms
*******************************************************************
version predcorr gam expon scale_data
HKM 1 0.000 1 0
it pstep dstep pinfeas dinfeas gap mean(obj) cputime
-------------------------------------------------------------------
0|0.000|0.000|4.7e+01|7.2e+00|1.2e+03| 2.500000e+01| 0:0:00| chol 1 1
1|0.736|0.733|1.2e+01|2.0e+00|3.9e+02| 1.289755e+01| 0:0:00| chol 1 1
2|0.830|0.798|2.1e+00|4.1e-01|1.1e+02| 1.352995e+01| 0:0:00| chol 1 1
3|0.788|0.831|4.4e-01|7.0e-02|3.2e+01| 9.062085e+00| 0:0:00| chol 1 1
4|0.732|1.000|1.2e-01|1.0e-04|1.1e+01| 5.408952e+00| 0:0:00| chol 1 1
5|0.914|1.000|1.0e-02|1.0e-05|1.0e+00| 5.088850e-01| 0:0:00| chol 1 1
6|1.000|1.000|5.5e-09|2.0e-03|4.9e-01| 2.816580e-01| 0:0:00| chol 1 1
7|0.873|1.000|1.8e-09|1.0e-07|6.4e-02| 9.353402e-02| 0:0:00| chol 1 1
8|1.000|0.864|2.3e-09|2.3e-08|1.6e-02| 8.394428e-02| 0:0:00| chol 1 1
9|0.967|0.939|3.9e-10|2.8e-09|7.8e-04| 7.891261e-02| 0:0:00| chol 1 1
10|0.970|0.966|1.2e-11|2.7e-10|2.5e-05| 7.868813e-02| 0:0:00| chol 1 1
11|1.000|1.000|1.2e-13|2.4e-12|1.6e-06| 7.868196e-02| 0:0:00| chol 1 1
12|1.000|1.000|1.7e-10|1.0e-12|1.6e-07| 7.868152e-02| 0:0:00| chol 1 1
13|1.000|1.000|5.0e-11|1.5e-12|3.0e-09| 7.868147e-02| 0:0:00|
stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
number of iterations = 13
primal objective value = 7.86814671e-02
dual objective value = 7.86814640e-02
gap := trace(XZ) = 3.04e-09
relative gap = 2.62e-09
actual relative gap = 2.62e-09
rel. primal infeas = 4.97e-11
rel. dual infeas = 1.50e-12
norm(X), norm(y), norm(Z) = 2.2e+00, 3.0e+00, 8.1e+00
norm(A), norm(b), norm(C) = 1.4e+01, 2.0e+00, 3.2e+00
Total CPU time (secs) = 0.3
CPU time per iteration = 0.0
termination code = 0
DIMACS: 5.0e-11 0.0e+00 2.4e-12 0.0e+00 2.6e-09 2.6e-09
-------------------------------------------------------------------
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.0786815