Quadratic discrimination (separating ellipsoid)
n = 2;
rand('state',0); randn('state',0);
N=50;
X = randn(2,N); X = X*diag(0.99*rand(1,N)./sqrt(sum(X.^2)));
Y = randn(2,N); Y = Y*diag((1.02+rand(1,N))./sqrt(sum(Y.^2)));
T = [1 -1; 2 1]; X = T*X; Y = T*Y;
fprintf(1,'Find the optimal ellipsoid that seperates the 2 classes...');
cvx_begin sdp
variable P(n,n) symmetric
variables q(n) r(1)
P <= -eye(n);
sum((X'*P).*X',2) + X'*q + r >= +1;
sum((Y'*P).*Y',2) + Y'*q + r <= -1;
cvx_end
fprintf(1,'Done! \n');
r = -r; P = -P; q = -q;
c = 0.25*q'*inv(P)*q - r;
xc = -0.5*inv(P)*q;
nopts = 1000;
angles = linspace(0,2*pi,nopts);
ell = inv(sqrtm(P/c))*[cos(angles); sin(angles)] + repmat(xc,1,nopts);
graph=plot(X(1,:),X(2,:),'o', Y(1,:), Y(2,:),'o', ell(1,:), ell(2,:),'-');
set(graph(2),'MarkerFaceColor',[0 0.5 0]);
set(gca,'XTick',[]); set(gca,'YTick',[]);
title('Quadratic discrimination');
Find the optimal ellipsoid that seperates the 2 classes...
Calling SeDuMi: 108 variables (5 free), 102 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
Split 5 free variables
eqs m = 102, order n = 113, dim = 115, blocks = 2
nnz(A) = 1206 + 0, nnz(ADA) = 10404, nnz(L) = 5253
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 2.17E-001 0.000
1 : 5.73E+001 1.04E-001 0.000 0.4809 0.9000 0.9000 -4.15 1 1 2.4E+001
2 : 5.84E+001 3.22E-002 0.000 0.3084 0.9000 0.9000 -2.53 1 1 6.3E+000
3 : 3.83E+001 7.39E-003 0.000 0.2296 0.9000 0.9000 -0.07 1 1 2.3E+000
4 : 2.16E+000 1.67E-004 0.000 0.0227 0.9900 0.9900 0.65 1 1 6.2E-002
5 : 9.15E-005 5.20E-009 0.000 0.0000 1.0000 1.0000 0.99 1 1 1.9E-006
6 : 2.42E-008 2.49E-012 0.000 0.0005 0.9999 0.9997 1.00 1 1 7.6E-010
iter seconds digits c*x b*y
6 0.1 Inf 0.0000000000e+000 2.4161599043e-008
|Ax-b| = 4.2e-009, [Ay-c]_+ = 7.4E-011, |x|= 4.0e+002, |y|= 1.1e-008
Detailed timing (sec)
Pre IPM Post
2.003E-002 5.007E-002 0.000E+000
Max-norms: ||b||=1, ||c|| = 0,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 18.1252.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0
Done!