Quadratic discrimination (separating ellipsoid)

% Section 8.6.2, Boyd & Vandenberghe "Convex Optimization"
% Original by Lieven Vandenberghe
% Adapted for CVX by Joelle Skaf - 10/16/05
% (a figure is generated)
%
% The goal is to find an ellipsoid that contains all the points
% x_1,...,x_N but none of the points y_1,...,y_M. The equation of the
% ellipsoidal surface is: z'*P*z + q'*z + r =0
% P, q and r can be obtained by solving the SDP feasibility problem:
%           minimize    0
%               s.t.    x_i'*P*x_i + q'*x_i + r >=  1   for i = 1,...,N
%                       y_i'*P*y_i + q'*y_i + r <= -1   for i = 1,...,M
%                       P <= -I

% data generation
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;

% Solution via CVX
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');

% Displaying results
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');
% print -deps ellips.eps
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!