Minimum volume ellipsoid covering a finite set
x = [ 0.55 0.0;
0.25 0.35
-0.2 0.2
-0.25 -0.1
-0.0 -0.3
0.4 -0.2 ]';
[n,m] = size(x);
cvx_begin
variable A(n,n) symmetric
variable b(n)
maximize( det_rootn( A ) )
subject to
norms( A * x + b * ones( 1, m ), 2 ) <= 1;
cvx_end
clf
noangles = 200;
angles = linspace( 0, 2 * pi, noangles );
ellipse = A \ [ cos(angles) - b(1) ; sin(angles) - b(2) ];
plot( x(1,:), x(2,:), 'ro', ellipse(1,:), ellipse(2,:), 'b-' );
axis off
Calling SeDuMi: 33 variables (2 free), 24 equality constraints
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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
Split 2 free variables
eqs m = 24, order n = 23, dim = 43, blocks = 9
nnz(A) = 72 + 0, nnz(ADA) = 354, nnz(L) = 189
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 8.47E-001 0.000
1 : 2.61E-001 2.06E-001 0.000 0.2434 0.9000 0.9000 2.99 1 1 5.6E-001
2 : -9.13E-001 6.93E-002 0.000 0.3360 0.9000 0.9000 0.76 1 1 2.6E-001
3 : -2.29E+000 1.56E-002 0.000 0.2252 0.9000 0.9000 0.42 1 1 7.6E-002
4 : -2.66E+000 1.54E-003 0.000 0.0985 0.9900 0.9900 0.91 1 1 7.7E-003
5 : -2.68E+000 2.96E-004 0.000 0.1927 0.9000 0.9000 1.00 1 1 1.5E-003
6 : -2.68E+000 9.15E-006 0.311 0.0309 0.9902 0.9900 1.00 1 1 9.2E-005
7 : -2.68E+000 1.04E-006 0.000 0.1139 0.9000 0.8650 1.00 1 1 1.9E-005
8 : -2.68E+000 2.94E-008 0.000 0.0282 0.9900 0.9763 1.00 1 1 5.3E-007
9 : -2.68E+000 1.26E-009 0.000 0.0429 0.9902 0.9900 1.00 2 2 2.3E-008
10 : -2.68E+000 7.90E-011 0.191 0.0627 0.9900 0.9529 1.00 2 2 1.5E-009
iter seconds digits c*x b*y
10 0.1 Inf -2.6839853947e+000 -2.6839853934e+000
|Ax-b| = 1.1e-009, [Ay-c]_+ = 7.6E-010, |x|= 1.1e+001, |y|= 2.5e+000
Detailed timing (sec)
Pre IPM Post
0.000E+000 1.102E-001 0.000E+000
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1145.68.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +2.68399