% Boyd & Vandenberghe "Convex Optimization"
% Joëlle Skaf - 09/26/05
%
% The 'fastest mixing Markov chain problem' is to find a transition
% probability matrix P on a graph E that minimizes the mixing rate r, where
% r = max{ lambda_2, -lambda_n } with lambda_1>=...>=lambda_n being the
% eigenvalues of P.

% Generate input data
n = 5;
E = [0 1 0 1 1; ...
     1 0 1 0 1; ...
     0 1 0 1 1; ...
     1 0 1 0 1; ...
     1 1 1 1 0];

% Create and solve model
cvx_begin
    variable P(n,n) symmetric
    minimize(norm(P - (1/n)*ones(n)))
    P*ones(n,1) == ones(n,1);
    P >= 0;
    P(E==0) == 0;
cvx_end
e = flipud(eig(P));
r = max(e(2), -e(n));

% Display results
disp('------------------------------------------------------------------------');
disp('The transition probability matrix of the optimal Markov chain is: ');
disp(P);
disp('The optimal mixing rate is: ');
disp(r);

 
Calling Mosek 9.1.9: 75 variables, 9 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            : 9               
  Cones                  : 0               
  Scalar variables       : 20              
  Matrix variables       : 1               
  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                   : CONIC (conic optimization problem)
  Constraints            : 9               
  Cones                  : 0               
  Scalar variables       : 20              
  Matrix variables       : 1               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 9
Optimizer  - Cones                  : 1
Optimizer  - Scalar variables       : 14                conic                  : 6               
Optimizer  - Semi-definite variables: 1                 scalarized             : 55              
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 : 45                after factor           : 45              
Factor     - dense dim.             : 0                 flops                  : 3.74e+03        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   9.0e+00  1.0e+00  1.0e+00  0.00e+00   0.000000000e+00   0.000000000e+00   1.0e+00  0.00  
1   3.0e+00  3.4e-01  1.2e-01  2.01e+00   -2.958763608e-01  -3.667314389e-01  3.4e-01  0.01  
2   9.8e-01  1.1e-01  2.5e-02  8.03e-01   -6.280645650e-01  -6.477150177e-01  1.1e-01  0.01  
3   5.5e-02  6.1e-03  3.0e-04  1.05e+00   -7.441472578e-01  -7.456864128e-01  6.1e-03  0.01  
4   2.5e-06  2.7e-07  6.1e-11  1.01e+00   -7.499999017e-01  -7.500000298e-01  2.7e-07  0.01  
5   6.2e-13  6.9e-14  1.5e-17  1.00e+00   -7.500000000e-01  -7.500000000e-01  6.9e-14  0.01  
Optimizer terminated. Time: 0.02    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -7.5000000000e-01   nrm: 1e+00    Viol.  con: 4e-13    var: 0e+00    barvar: 0e+00  
  Dual.    obj: -7.5000000000e-01   nrm: 8e-01    Viol.  con: 0e+00    var: 9e-15    barvar: 5e-14  
Optimizer summary
  Optimizer                 -                        time: 0.02    
    Interior-point          - iterations : 5         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): +0.75
 
------------------------------------------------------------------------
The transition probability matrix of the optimal Markov chain is: 
         0    0.3750         0    0.3750    0.2500
    0.3750         0    0.3750         0    0.2500
         0    0.3750         0    0.3750    0.2500
    0.3750         0    0.3750         0    0.2500
    0.2500    0.2500    0.2500    0.2500         0

The optimal mixing rate is: 
    0.7500