% 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 points
% {x_1,...,x_N} and {y_1,...,y_M} with maximal 'gap'. a and b can be
% obtained by solving the following problem:
%           maximize    t
%               s.t.    a'*x_i - b >=  t     for i = 1,...,N
%                       a'*y_i - b <= -t     for i = 1,...,M
%                       ||a||_2 <= 1

% data generation
n = 2;
randn('state',3);
N = 10; M = 6;
Y = [1.5+1*randn(1,M); 2*randn(1,M)];
X = [-1.5+1*randn(1,N); 2*randn(1,N)];
T = [-1 1; 1 1];
Y = T*Y;  X = T*X;

% Solution via CVX
cvx_begin
    variables a(n) b(1) t(1)
    maximize (t)
    X'*a - b >= t;
    Y'*a - b <= -t;
    norm(a) <= 1;
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+t)/a(2);
p2 = -a(1)*tt/a(2) + (b-t)/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('Robust linear discrimination problem');
% print -deps linsep.eps

 
Calling Mosek 9.1.9: 20 variables, 5 equality constraints
   For improved efficiency, Mosek is solving the dual problem.
------------------------------------------------------------

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            : 5               
  Cones                  : 1               
  Scalar variables       : 20              
  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            : 5               
  Cones                  : 1               
  Scalar variables       : 20              
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 4
Optimizer  - Cones                  : 1
Optimizer  - Scalar variables       : 19                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 : 10                after factor           : 10              
Factor     - dense dim.             : 0                 flops                  : 3.60e+02        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   1.0e+00  0.0e+00  2.0e+00  0.00e+00   1.000000000e+00   0.000000000e+00   1.0e+00  0.00  
1   2.4e-01  2.2e-16  1.0e-01  1.75e+00   4.928262446e-02   -2.162170645e-01  2.4e-01  0.01  
2   6.4e-02  8.9e-16  3.3e-02  9.56e-01   7.539605392e-02   -7.248178782e-03  6.4e-02  0.01  
3   3.3e-02  4.4e-16  1.8e-02  -1.71e-01  1.780038238e-01   1.362995556e-01   3.3e-02  0.01  
4   1.4e-02  8.9e-16  2.9e-03  1.16e+00   4.473413906e-01   4.211110196e-01   1.4e-02  0.01  
5   2.0e-03  8.9e-16  1.8e-04  7.73e-01   5.002993976e-01   4.962765005e-01   2.0e-03  0.01  
6   2.1e-05  4.0e-15  2.0e-07  9.67e-01   5.111203252e-01   5.110796674e-01   2.1e-05  0.01  
7   6.6e-07  7.1e-15  1.1e-09  1.00e+00   5.112279286e-01   5.112266697e-01   6.6e-07  0.01  
8   9.0e-08  1.2e-14  5.7e-11  1.00e+00   5.112297451e-01   5.112295759e-01   9.0e-08  0.01  
9   6.4e-09  9.2e-14  1.1e-12  1.00e+00   5.112298887e-01   5.112298772e-01   6.4e-09  0.01  
Optimizer terminated. Time: 0.01    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 5.1122988874e-01    nrm: 1e+00    Viol.  con: 9e-09    var: 5e-09    cones: 0e+00  
  Dual.    obj: 5.1122987722e-01    nrm: 6e+00    Viol.  con: 0e+00    var: 1e-13    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.01    
    Interior-point          - iterations : 9         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): +0.51123