% 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 SDPT3 4.0: 103 variables, 6 equality constraints
   For improved efficiency, SDPT3 is solving the dual problem.
------------------------------------------------------------

 num. of constraints =  6
 dim. of sdp    var  =  2,   num. of sdp  blk  =  1
 dim. of linear var  = 100
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
   HKM      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
 0|0.000|0.000|2.9e+03|9.6e+00|6.4e+04| 2.626414e+03  0.000000e+00| 0:0:00| chol  1  1 
 1|0.636|0.827|1.0e+03|1.7e+00|3.0e+04| 7.649394e+02  0.000000e+00| 0:0:00| chol  1  1 
 2|0.797|1.000|2.1e+02|1.6e-03|6.7e+03| 1.293218e+02  0.000000e+00| 0:0:00| chol  1  1 
 3|0.987|1.000|2.7e+00|1.6e-04|8.4e+01| 1.627774e+00  0.000000e+00| 0:0:00| chol  1  1 
 4|0.989|1.000|2.9e-02|1.6e-05|9.3e-01| 1.791336e-02  0.000000e+00| 0:0:00| chol  1  1 
 5|0.989|1.000|3.2e-04|5.9e-03|1.0e-02| 1.968772e-04  0.000000e+00| 0:0:00| chol  1  1 
 6|0.989|1.000|3.6e-06|6.5e-05|1.1e-04| 2.176872e-06  0.000000e+00| 0:0:00| chol  1  1 
 7|0.989|1.000|3.9e-08|7.3e-07|1.2e-06| 2.392715e-08  0.000000e+00| 0:0:00| chol  1  1 
 8|0.983|1.000|6.9e-10|7.9e-09|2.2e-08| 4.410963e-10  0.000000e+00| 0:0:00| chol  1  1 
 9|0.985|1.000|1.1e-11|1.4e-10|3.4e-10| 6.901849e-12  0.000000e+00| 0:0:00|
  stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
 number of iterations   =  9
 primal objective value =  6.90184871e-12
 dual   objective value =  0.00000000e+00
 gap := trace(XZ)       = 3.40e-10
 relative gap           = 3.40e-10
 actual relative gap    = 6.90e-12
 rel. primal infeas (scaled problem)   = 1.06e-11
 rel. dual     "        "       "      = 1.38e-10
 rel. primal infeas (unscaled problem) = 0.00e+00
 rel. dual     "        "       "      = 0.00e+00
 norm(X), norm(y), norm(Z) = 9.9e-13, 1.2e+02, 1.1e+03
 norm(A), norm(b), norm(C) = 7.8e+01, 1.0e+00, 6.4e+01
 Total CPU time (secs)  = 0.13  
 CPU time per iteration = 0.01  
 termination code       =  0
 DIMACS: 1.1e-11  0.0e+00  4.5e-10  0.0e+00  6.9e-12  3.4e-10
-------------------------------------------------------------------
 
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): -6.90185e-12
 
Done!