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: 103 variables, 6 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 = 6, order n = 103, dim = 105, blocks = 2
nnz(A) = 603 + 0, nnz(ADA) = 36, nnz(L) = 21
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 3.47E+02 0.000
1 : 0.00E+00 1.17E+02 0.000 0.3363 0.9000 0.9000 -3.81 1 1 1.2E+02
2 : 0.00E+00 3.46E+01 0.000 0.2965 0.9000 0.9000 -0.20 1 1 6.3E+01
3 : 0.00E+00 1.67E+00 0.000 0.0482 0.9900 0.9900 0.39 1 1 4.2E+00
4 : 0.00E+00 1.09E-03 0.086 0.0007 0.9999 0.9999 0.96 1 1 2.8E-03
5 : 0.00E+00 1.30E-05 0.323 0.0120 0.9945 0.9945 1.00 1 1 3.4E-05
6 : 0.00E+00 2.77E-07 0.000 0.0212 0.9900 0.9792 1.00 1 1 7.3E-07
7 : 0.00E+00 7.51E-14 0.000 0.0000 1.0000 1.0000 1.00 1 1 2.0E-13
iter seconds digits c*x b*y
7 0.0 Inf -5.6712804946e-14 0.0000000000e+00
|Ax-b| = 1.7e-13, [Ay-c]_+ = 0.0E+00, |x|= 3.4e-14, |y|= 1.8e+01
Detailed timing (sec)
Pre IPM Post
0.000E+00 2.000E-02 1.000E-02
Max-norms: ||b||=0, ||c|| = 7.058515e+00,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.9834.
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Status: Solved
Optimal value (cvx_optval): +5.67128e-14
Done!