% Boyd & Vandenberghe, "Convex Optimization"
% Joëlle Skaf - 08/16/05
% (a figure is generated)
%
% The goal is to find the largest Euclidean ball (i.e. its center and
% radius) that lies in a polyhedron described by linear inequalites in this
% fashion: P = {x : a_i'*x <= b_i, i=1,...,m} where x is in R^2

% Generate the input data
a1 = [ 2;  1];
a2 = [ 2; -1];
a3 = [-1;  2];
a4 = [-1; -2];
b = ones(4,1);

% Create and solve the model
cvx_begin
    variable r(1)
    variable x_c(2)
    maximize ( r )
    a1'*x_c + r*norm(a1,2) <= b(1);
    a2'*x_c + r*norm(a2,2) <= b(2);
    a3'*x_c + r*norm(a3,2) <= b(3);
    a4'*x_c + r*norm(a4,2) <= b(4);
cvx_end

% Generate the figure
x = linspace(-2,2);
theta = 0:pi/100:2*pi;
plot( x, -x*a1(1)./a1(2) + b(1)./a1(2),'b-');
hold on
plot( x, -x*a2(1)./a2(2) + b(2)./a2(2),'b-');
plot( x, -x*a3(1)./a3(2) + b(3)./a3(2),'b-');
plot( x, -x*a4(1)./a4(2) + b(4)./a4(2),'b-');
plot( x_c(1) + r*cos(theta), x_c(2) + r*sin(theta), 'r');
plot(x_c(1),x_c(2),'k+')
xlabel('x_1')
ylabel('x_2')
title('Largest Euclidean ball lying in a 2D polyhedron');
axis([-1 1 -1 1])
axis equal

 
Calling Mosek 9.1.9: 4 variables, 3 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                   : LO (linear optimization problem)
  Constraints            : 3               
  Cones                  : 0               
  Scalar variables       : 4               
  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 started.
Freed constraints in eliminator : 0
Eliminator terminated.
Eliminator - tries                  : 2                 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            : 3               
  Cones                  : 0               
  Scalar variables       : 4               
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 3
Optimizer  - Cones                  : 0
Optimizer  - Scalar variables       : 4                 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 : 6                 after factor           : 6               
Factor     - dense dim.             : 0                 flops                  : 6.20e+01        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   6.0e+00  0.0e+00  8.5e+00  0.00e+00   4.000000000e+00   0.000000000e+00   2.9e+00  0.00  
1   1.3e+00  4.4e-16  1.9e+00  1.74e+00   9.333165808e-01   2.492631633e-01   6.4e-01  0.01  
2   3.5e-02  8.9e-16  5.0e-02  3.09e+00   4.532518544e-01   4.456360007e-01   1.8e-02  0.01  
3   4.8e-06  8.9e-16  6.9e-06  1.02e+00   4.472144123e-01   4.472134243e-01   2.5e-06  0.01  
4   4.8e-10  8.9e-16  6.9e-10  1.00e+00   4.472135956e-01   4.472135955e-01   2.5e-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: 4.4721359558e-01    nrm: 1e+00    Viol.  con: 2e-10    var: 0e+00  
  Dual.    obj: 4.4721359548e-01    nrm: 4e-01    Viol.  con: 0e+00    var: 2e-18  

Basic solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: 4.4721359550e-01    nrm: 1e+00    Viol.  con: 1e-17    var: 0e+00  
  Dual.    obj: 4.4721359548e-01    nrm: 4e-01    Viol.  con: 0e+00    var: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.02    
    Interior-point          - iterations : 4         time: 0.01    
      Basis identification  -                        time: 0.00    
        Primal              - iterations : 1         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): +0.447214