Figure 8.10: Approximate linear discrimination via linear programming

% Section 8.6.1, 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 a function f(x) = a'*x - b that classifies the non-
% separable points {x_1,...,x_N} and {y_1,...,y_M} by allowing some
% misclassification. a and b can be obtained by solving the following
% problem:
%           minimize    1'*u + 1'*v
%               s.t.    a'*x_i - b >= 1 - u_i        for i = 1,...,N
%                       a'*y_i - b <= -(1 - v_i)     for i = 1,...,M
%                       u >= 0 and v >= 0

% data generation
n = 2;
randn('state',2);
N = 50; M = 50;
Y = [1.5+0.9*randn(1,0.6*N), 1.5+0.7*randn(1,0.4*N);
     2*(randn(1,0.6*N)+1), 2*(randn(1,0.4*N)-1)];
X = [-1.5+0.9*randn(1,0.6*M),  -1.5+0.7*randn(1,0.4*M);
      2*(randn(1,0.6*M)-1), 2*(randn(1,0.4*M)+1)];
T = [-1 1; 1 1];
Y = T*Y;  X = T*X;

% Solution via CVX
cvx_begin
    variables a(n) b(1) u(N) v(M)
    minimize (ones(1,N)*u + ones(1,M)*v)
    X'*a - b >= 1 - u;
    Y'*a - b <= -(1 - v);
    u >= 0;
    v >= 0;
cvx_end

% Displaying results
linewidth = 0.5;  % for the squares and circles
t_min = min([X(1,:),Y(1,:)]);
t_max = max([X(1,:),Y(1,:)]);
tt = linspace(t_min-1,t_max+1,100);
p = -a(1)*tt/a(2) + b/a(2);
p1 = -a(1)*tt/a(2) + (b+1)/a(2);
p2 = -a(1)*tt/a(2) + (b-1)/a(2);

graph = plot(X(1,:),X(2,:), 'o', Y(1,:), Y(2,:), 'o');
set(graph(1),'LineWidth',linewidth);
set(graph(2),'LineWidth',linewidth);
set(graph(2),'MarkerFaceColor',[0 0.5 0]);
hold on;
plot(tt,p, '-r', tt,p1, '--r', tt,p2, '--r');
axis equal
title('Approximate linear discrimination via linear programming');
% print -deps svc-discr.eps
 
Calling SeDuMi: 203 variables (3 free), 100 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 3 free variables
eqs m = 100, order n = 207, dim = 207, blocks = 1
nnz(A) = 200 + 600, nnz(ADA) = 100, nnz(L) = 100
Handling 6 + 0 dense columns.
 it :     b*y       gap    delta  rate   t/tP*  t/tD*   feas cg cg  prec
  0 :            5.18E+001 0.000
  1 :  1.53E+001 2.32E+001 0.000 0.4484 0.9000 0.9000   2.92  1  1  1.4E+000
  2 :  1.38E+001 6.94E+000 0.000 0.2989 0.9000 0.9000   1.34  1  1  5.7E-001
  3 :  1.12E+001 2.10E+000 0.000 0.3032 0.9000 0.9000   0.87  1  1  3.0E-001
  4 :  1.03E+001 1.05E+000 0.000 0.4984 0.9000 0.9000   0.79  1  1  2.1E-001
  5 :  9.94E+000 6.88E-001 0.000 0.6561 0.9000 0.9000   0.68  1  1  1.8E-001
  6 :  9.50E+000 2.38E-001 0.000 0.3459 0.9000 0.9000   0.37  1  1  1.8E-001
  7 :  8.22E+000 1.34E-001 0.000 0.5623 0.9000 0.9000   0.13  1  1  1.3E-001
  8 :  7.83E+000 8.96E-002 0.000 0.6698 0.9000 0.9000   0.62  1  1  1.2E-001
  9 :  7.65E+000 7.10E-002 0.000 0.7920 0.9000 0.9000   0.27  1  1  1.2E-001
 10 :  6.79E+000 3.11E-002 0.000 0.4378 0.9000 0.9000   0.79  1  1  6.2E-002
 11 :  6.47E+000 1.43E-002 0.000 0.4609 0.9000 0.9000   0.85  1  1  3.3E-002
 12 :  6.36E+000 7.87E-003 0.000 0.5494 0.9000 0.9000   0.83  1  1  2.2E-002
 13 :  6.21E+000 2.33E-003 0.000 0.2957 0.9000 0.9000   0.93  1  1  6.6E-003
 14 :  6.15E+000 1.05E-004 0.000 0.0452 0.9900 0.9900   0.96  1  1  
iter seconds digits       c*x               b*y
 14      0.1  14.8  6.1485694323e+000  6.1485694323e+000
|Ax-b| =  9.8e-014, [Ay-c]_+ =  2.1E-015, |x|= 9.0e+001, |y|= 2.4e+000

Detailed timing (sec)
   Pre          IPM          Post
0.000E+000    1.102E-001    0.000E+000    
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +6.14857