% Section 11.4.1, Boyd & Vandenberghe "Convex Optimization"
% Written for CVX by Almir Mutapcic - 02/18/06
%
% We consider a set of linear inequalities A*x <= b which are
% infeasible. Here A is a matrix in R^(m-by-n) and b belongs
% to R^m. We apply a heuristic to find a point x that violates
% only a small number of inequalities.
%
% We use the sum of infeasibilities heuristic:
%
%   minimize   sum( max( Ax - b ) )
%
% which is equivalent to the following LP (book pg. 580):
%
%   minimize   sum( s )
%       s.t.   Ax <= b + s
%              s >= 0
%
% with variables x in R^n and s in R^m.

% problem dimensions (m inequalities in n-dimensional space)
m = 150;
n = 10;

% fix random number generator so we can repeat the experiment
seed = 0;
randn('state',seed);

% construct infeasible inequalities
A = randn(m,n);
b = randn(m,1);

fprintf(1, ['Starting with an infeasible set of %d inequalities ' ...
            'in %d variables.\n'],m,n);

% sum of infeasibilities heuristic
cvx_begin
   variable x(n)
   minimize( sum( max( A*x - b, 0 ) ) )
cvx_end

% full LP version of the sum of infeasibilities heuristic
% cvx_begin
%   variables x(n) s(m)
%   minimize( sum( s ) )
%   subject to
%     A*x <= b + s;
%     s >= 0;
% cvx_end

% number of satisfied inequalities
nv = length( find( A*x > b ) );
fprintf(1,'\nFound an x that violates %d out of %d inequalities.\n',nv,m);

Starting with an infeasible set of 150 inequalities in 10 variables.
 
Calling Mosek 9.1.9: 310 variables, 150 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                   : LO (linear optimization problem)
  Constraints            : 150             
  Cones                  : 0               
  Scalar variables       : 310             
  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                   : LO (linear optimization problem)
  Constraints            : 150             
  Cones                  : 0               
  Scalar variables       : 310             
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the dual        
Optimizer  - Constraints            : 10
Optimizer  - Cones                  : 0
Optimizer  - Scalar variables       : 150               conic                  : 0               
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 : 55                after factor           : 55              
Factor     - dense dim.             : 0                 flops                  : 1.69e+04        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   1.2e+01  2.8e+01  9.5e+01  0.00e+00   0.000000000e+00   -3.226946936e+01  8.1e+00  0.00  
1   2.2e+00  5.4e+00  1.8e+01  8.52e-01   3.172284918e+01   2.579425457e+01   1.6e+00  0.01  
2   3.6e-01  8.7e-01  3.0e+00  1.04e+00   3.721624826e+01   3.626522466e+01   2.5e-01  0.01  
3   7.2e-02  1.7e-01  5.9e-01  9.93e-01   3.855781259e+01   3.836666385e+01   5.1e-02  0.01  
4   8.0e-03  1.9e-02  6.5e-02  9.96e-01   3.887732250e+01   3.885616107e+01   5.6e-03  0.01  
5   9.6e-04  2.3e-03  7.9e-03  1.00e+00   3.891202591e+01   3.890947834e+01   6.7e-04  0.01  
6   6.3e-05  1.5e-04  5.1e-04  1.00e+00   3.891634738e+01   3.891618213e+01   4.4e-05  0.01  
7   1.9e-06  4.5e-06  1.5e-05  1.00e+00   3.891675726e+01   3.891675238e+01   1.3e-06  0.01  
8   1.9e-10  4.6e-10  1.6e-09  1.00e+00   3.891676296e+01   3.891676296e+01   1.3e-10  0.01  
Basis identification started.
Primal basis identification phase started.
Primal basis identification phase terminated. Time: 0.00
Dual basis identification phase started.
Dual basis identification phase terminated. Time: 0.00
Basis identification terminated. Time: 0.00
Optimizer terminated. Time: 0.02    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 3.8916762961e+01    nrm: 3e+00    Viol.  con: 5e-11    var: 0e+00  
  Dual.    obj: 3.8916762958e+01    nrm: 1e+00    Viol.  con: 0e+00    var: 4e-10  

Basic solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 3.8916762954e+01    nrm: 3e+00    Viol.  con: 5e-09    var: 0e+00  
  Dual.    obj: 3.8916762959e+01    nrm: 1e+00    Viol.  con: 0e+00    var: 3e-15  
Optimizer summary
  Optimizer                 -                        time: 0.02    
    Interior-point          - iterations : 8         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): +38.9168
 

Found an x that violates 57 out of 150 inequalities.