# Quadratic discrimination (separating ellipsoid)

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    % 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: 103 variables, 6 equality constraints
       For improved efficiency, sedumi is solving the dual problem.
    ------------------------------------------------------------
    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 :            3.28E+02 0.000
      1 :   0.00E+00 1.15E+02 0.000 0.3515 0.9000 0.9000  -1.39  1  1  2.9E+02
      2 :   0.00E+00 2.31E+01 0.000 0.2005 0.9000 0.9000  -0.10  1  1  9.8E+01
      3 :   0.00E+00 6.60E-01 0.000 0.0286 0.9900 0.9900   0.59  1  1  6.5E+00
      4 :   0.00E+00 5.06E-05 0.000 0.0001 1.0000 1.0000   0.99  1  1  1.3E-02
      5 :   0.00E+00 5.25E-12 0.000 0.0000 1.0000 1.0000   1.00  1  1  2.9E-09

    iter seconds digits       c*x               b*y
      5      0.0   2.6  3.2204638088e-13  0.0000000000e+00
    |Ax-b| =   1.5e-11, [Ay-c]_+ =   0.0E+00, |x|=  6.1e-12, |y|=  7.7e+01

    Detailed timing (sec)
       Pre          IPM          Post
    1.000E-02    2.000E-02    0.000E+00
    Max-norms: ||b||=0, ||c|| = 1.845499e+01,
    Cholesky |add|=0, |skip| = 0, ||L.L|| = 3.37316.
    ------------------------------------------------------------
    Status: Solved
    Optimal value (cvx_optval): -3.22046e-13
    Done!