% 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 doing a trade-off
% between the number of misclassifications and the width of the separating
% slab. a and b can be obtained by solving the following problem:
%           minimize    ||a||_2 + gamma*(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
% where gamma gives the relative weight of the number of misclassified
% points compared to the width of the slab.

% 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;
g = 0.1;            % gamma

% Solution via CVX
cvx_begin
    variables a(n) b(1) u(N) v(M)
    minimize (norm(a) + g*(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 support vector classifier');
% print -deps svc-discr2.eps

 
Calling Mosek 9.1.9: 204 variables, 100 equality constraints
------------------------------------------------------------

MOSEK Version 9.1.9 (Build date: 2019-11-21 11:32:15)
Copyright (c) MOSEK ApS, Denmark. WWW: mosek.com
Platform: MACOSX/64-X86

Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 100             
  Cones                  : 1               
  Scalar variables       : 204             
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 1                 time                   : 0.00            
Lin. dep.  - tries                  : 1                 time                   : 0.00            
Lin. dep.  - number                 : 0               
Presolve terminated. Time: 0.00    
Problem
  Name                   :                 
  Objective sense        : min             
  Type                   : CONIC (conic optimization problem)
  Constraints            : 100             
  Cones                  : 1               
  Scalar variables       : 204             
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the dual        
Optimizer  - Constraints            : 3
Optimizer  - Cones                  : 1
Optimizer  - Scalar variables       : 103               conic                  : 3               
Optimizer  - Semi-definite variables: 0                 scalarized             : 0               
Factor     - setup time             : 0.00              dense det. time        : 0.00            
Factor     - ML order time          : 0.00              GP order time          : 0.00            
Factor     - nonzeros before factor : 6                 after factor           : 6               
Factor     - dense dim.             : 0                 flops                  : 1.22e+03        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   1.0e+00  1.0e+00  2.0e+00  0.00e+00   1.000000000e+00   0.000000000e+00   1.0e+00  0.00  
1   5.4e-01  5.4e-01  5.0e-01  7.32e-01   4.022341411e+00   3.352701347e+00   5.4e-01  0.01  
2   1.8e-01  1.8e-01  4.2e-02  1.93e+00   3.625762002e+00   3.482760180e+00   1.8e-01  0.01  
3   6.8e-02  6.8e-02  1.2e-02  1.36e+00   2.712319677e+00   2.662292697e+00   6.8e-02  0.01  
4   2.7e-02  2.7e-02  3.4e-03  8.62e-01   2.288778881e+00   2.267268446e+00   2.7e-02  0.01  
5   1.5e-02  1.5e-02  1.5e-03  7.41e-01   2.115194629e+00   2.102340577e+00   1.5e-02  0.01  
6   6.5e-03  6.5e-03  4.8e-04  8.05e-01   1.961321530e+00   1.955476117e+00   6.5e-03  0.01  
7   3.4e-03  3.4e-03  1.9e-04  9.15e-01   1.902225605e+00   1.899088083e+00   3.4e-03  0.01  
8   1.1e-03  1.1e-03  3.7e-05  9.36e-01   1.852267765e+00   1.851266681e+00   1.1e-03  0.01  
9   4.9e-04  4.9e-04  1.1e-05  9.34e-01   1.838260383e+00   1.837814862e+00   4.9e-04  0.01  
10  3.4e-05  3.4e-05  2.3e-07  9.96e-01   1.826707776e+00   1.826678930e+00   3.4e-05  0.01  
11  4.8e-06  4.8e-06  1.3e-08  9.98e-01   1.825838396e+00   1.825834275e+00   4.8e-06  0.01  
12  1.1e-06  1.1e-06  1.5e-09  1.00e+00   1.825734335e+00   1.825733365e+00   1.1e-06  0.01  
13  3.0e-07  3.0e-07  2.0e-10  1.00e+00   1.825709119e+00   1.825708865e+00   3.0e-07  0.01  
14  4.3e-09  4.3e-09  3.3e-13  1.00e+00   1.825700346e+00   1.825700342e+00   4.3e-09  0.01  
Optimizer terminated. Time: 0.02    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 1.8257003460e+00    nrm: 3e+00    Viol.  con: 7e-16    var: 0e+00    cones: 0e+00  
  Dual.    obj: 1.8257003425e+00    nrm: 1e+00    Viol.  con: 0e+00    var: 2e-09    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.02    
    Interior-point          - iterations : 14        time: 0.01    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 0         time: 0.00    
        Dual                - iterations : 0         time: 0.00    
        Clean primal        - iterations : 0         time: 0.00    
        Clean dual          - iterations : 0         time: 0.00    
    Simplex                 -                        time: 0.00    
      Primal simplex        - iterations : 0         time: 0.00    
      Dual simplex          - iterations : 0         time: 0.00    
    Mixed integer           - relaxations: 0         time: 0.00    

------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +1.8257