% Section 7.2, Figures 7.2-7.3
% Boyd & Vandenberghe, "Convex Optimization"
% Originally by Lieven Vandenberghe
% Adapted for CVX by Michael Grant 4/11/06
%
% We consider a probability distribution on 100 equidistant points in the
% interval [-1,1]. We impose the following prior assumptions:
%
%    -0.1 <= E(X) <= +0.1
%    +0.5 <= E(X^2) <= +0.6
%    -0.3 <= E(3*X^3-2*X) <= -0.2
%    +0.3 <= Pr(X<0) <= 0.4
%
% Along with the constraints sum(p) == 1, p >= 0, these constraints
% describe a polyhedron of probability distrubtions. In the first figure,
% the distribution that maximizes entropy is computed. In the second
% figure, we compute upper and lower bounds for Prob(X<=a_i) for each
% point -1 <= a_i <= +1 in the distribution, as well as the maximum
% entropy CDF.

%
% Represent the polyhedron as follows:
%     A * p <= b
%     sum( p ) == 1
%     p >= 0
%

n  = 100;
a  = linspace(-1,1,n);
a2 = a .^ 2;
a3 = 3 * ( a.^ 3 ) - 2 * a;
ap = +( a < 0 );
A  = [ a   ; -a  ; a2 ; -a2  ; a3 ; -a3 ; ap ; -ap ];
b  = [ 0.1 ; 0.1 ;0.5 ; -0.5 ; -0.2 ; 0.3 ; 0.4 ; -0.3 ];

%
% Compute the maximum entropy distribution
%

cvx_expert true
cvx_begin
    variables pent(n)
    maximize( sum(entr(pent)) )
    sum(pent) == 1;
    A * pent <= b;
cvx_end

%
% Compute the bounds on Prob(X<=a_i), i=1,...,n
%

Ubnds = zeros(1,n);
Lbnds = zeros(1,n);
for t = 1 : n,
    cvx_begin quiet
        variable p( n )
        minimize sum( p(1:t) )
        p >= 0;
        sum( p ) == 1;
        A * p <= b;
    cvx_end
    Lbnds(t) = cvx_optval;
    cvx_begin quiet
        variable p( n )
        maximize sum( p(1:t) )
        p >= 0;
        sum( p ) == 1;
        A * p <= b;
    cvx_end
    Ubnds(t) = cvx_optval;
    disp( sprintf( '%g <= Prob(x<=%g) <= %g', Lbnds(t), a(t), Ubnds(t) ) );
end

%
% Generate the figures
%

figure( 1 )
stairs( a, pent );
xlabel( 'x' );
ylabel( 'PDF( x )' );

figure( 2 )
stairs( a, cumsum( pent ) );
grid on;
hold on
d = stairs(a, Lbnds,'r-');  set(d,'Color',[0 0.5 0]);
d = stairs(a, Ubnds,'r-');  set(d,'Color',[0 0.5 0]);
d = plot([-1,-1], [Lbnds(1), Ubnds(1)],'r-');
set(d,'Color',[0 0.5 0]);
axis([-1.1 1.1 -0.1 1.1]);
xlabel( 'x' );
ylabel( 'CDF( x )' );
hold off

 
Calling Mosek 9.1.9: 308 variables, 109 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            : 109             
  Cones                  : 100             
  Scalar variables       : 308             
  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            : 109             
  Cones                  : 100             
  Scalar variables       : 308             
  Matrix variables       : 0               
  Integer variables      : 0               

Optimizer  - threads                : 8               
Optimizer  - solved problem         : the primal      
Optimizer  - Constraints            : 5
Optimizer  - Cones                  : 100
Optimizer  - Scalar variables       : 303               conic                  : 300             
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 : 15                after factor           : 15              
Factor     - dense dim.             : 0                 flops                  : 5.06e+03        
ITE PFEAS    DFEAS    GFEAS    PRSTATUS   POBJ              DOBJ              MU       TIME  
0   8.0e+01  8.1e-01  2.1e+02  0.00e+00   8.278383991e+01   -1.290927710e+02  1.0e+00  0.00  
1   9.2e+00  9.3e-02  1.2e+01  6.35e-01   -1.258106333e+01  -4.378494190e+01  1.2e-01  0.01  
2   1.1e+00  1.2e-02  8.5e-01  5.78e-01   -7.925074971e+00  -1.308368634e+01  1.4e-02  0.01  
3   1.9e-01  1.9e-03  6.5e-02  7.63e-01   -4.920183301e+00  -5.898967691e+00  2.4e-03  0.01  
4   1.9e-02  1.9e-04  1.6e-03  1.14e+00   -4.426324685e+00  -4.514412441e+00  2.4e-04  0.01  
5   1.7e-03  1.7e-05  3.9e-05  1.05e+00   -4.389622164e+00  -4.397222024e+00  2.1e-05  0.01  
6   9.8e-05  9.9e-07  5.4e-07  1.01e+00   -4.386539903e+00  -4.386981889e+00  1.2e-06  0.01  
7   9.9e-06  1.0e-07  1.7e-08  1.00e+00   -4.386323068e+00  -4.386367836e+00  1.2e-07  0.01  
8   1.3e-06  1.3e-08  8.4e-10  1.00e+00   -4.386298489e+00  -4.386304439e+00  1.7e-08  0.01  
9   1.8e-07  1.8e-09  4.1e-11  1.00e+00   -4.386294908e+00  -4.386295698e+00  2.2e-09  0.01  
10  5.3e-08  5.3e-10  6.7e-12  1.00e+00   -4.386294501e+00  -4.386294738e+00  6.6e-10  0.01  
11  3.2e-08  3.2e-10  3.1e-12  9.99e-01   -4.386294437e+00  -4.386294581e+00  4.0e-10  0.01  
12  3.2e-08  3.2e-10  3.1e-12  1.08e+00   -4.386294436e+00  -4.386294578e+00  4.0e-10  0.01  
13  3.1e-08  3.2e-10  3.0e-12  9.85e-01   -4.386294435e+00  -4.386294575e+00  3.9e-10  0.02  
14  3.1e-08  3.2e-10  3.0e-12  9.97e-01   -4.386294435e+00  -4.386294575e+00  3.9e-10  0.02  
15  7.6e-09  7.9e-11  3.5e-13  9.99e-01   -4.386294359e+00  -4.386294394e+00  9.7e-11  0.02  
Optimizer terminated. Time: 0.02    


Interior-point solution summary
  Problem status  : PRIMAL_AND_DUAL_FEASIBLE
  Solution status : OPTIMAL
  Primal.  obj: -4.3862943588e+00   nrm: 1e+00    Viol.  con: 1e-08    var: 0e+00    cones: 0e+00  
  Dual.    obj: -4.3862943940e+00   nrm: 5e+00    Viol.  con: 0e+00    var: 6e-13    cones: 0e+00  
Optimizer summary
  Optimizer                 -                        time: 0.02    
    Interior-point          - iterations : 15        time: 0.02    
      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): +4.38629
 
0 <= Prob(x<=-1) <= 0.329406
0 <= Prob(x<=-0.979798) <= 0.344777
0 <= Prob(x<=-0.959596) <= 0.360771
0 <= Prob(x<=-0.939394) <= 0.377365
0 <= Prob(x<=-0.919192) <= 0.394532
0 <= Prob(x<=-0.89899) <= 0.4
0 <= Prob(x<=-0.878788) <= 0.4
0 <= Prob(x<=-0.858586) <= 0.4
0 <= Prob(x<=-0.838384) <= 0.4
0 <= Prob(x<=-0.818182) <= 0.4
0 <= Prob(x<=-0.79798) <= 0.4
0.0116304 <= Prob(x<=-0.777778) <= 0.4
0.0331953 <= Prob(x<=-0.757576) <= 0.4
0.0519157 <= Prob(x<=-0.737374) <= 0.4
0.0701091 <= Prob(x<=-0.717172) <= 0.4
0.0859495 <= Prob(x<=-0.69697) <= 0.4
0.0998612 <= Prob(x<=-0.676768) <= 0.4
0.112141 <= Prob(x<=-0.656566) <= 0.4
0.123045 <= Prob(x<=-0.636364) <= 0.4
0.132778 <= Prob(x<=-0.616162) <= 0.4
0.141527 <= Prob(x<=-0.59596) <= 0.4
0.149418 <= Prob(x<=-0.575758) <= 0.4
0.15655 <= Prob(x<=-0.555556) <= 0.4
0.163015 <= Prob(x<=-0.535354) <= 0.4
0.168895 <= Prob(x<=-0.515152) <= 0.4
0.174283 <= Prob(x<=-0.494949) <= 0.4
0.179205 <= Prob(x<=-0.474747) <= 0.4
0.18371 <= Prob(x<=-0.454545) <= 0.4
0.187841 <= Prob(x<=-0.434343) <= 0.4
0.191651 <= Prob(x<=-0.414141) <= 0.4
0.195164 <= Prob(x<=-0.393939) <= 0.4
0.198396 <= Prob(x<=-0.373737) <= 0.4
0.201373 <= Prob(x<=-0.353535) <= 0.4
0.204127 <= Prob(x<=-0.333333) <= 0.4
0.206681 <= Prob(x<=-0.313131) <= 0.4
0.209037 <= Prob(x<=-0.292929) <= 0.4
0.211209 <= Prob(x<=-0.272727) <= 0.4
0.213219 <= Prob(x<=-0.252525) <= 0.4
0.215088 <= Prob(x<=-0.232323) <= 0.4
0.216811 <= Prob(x<=-0.212121) <= 0.4
0.218398 <= Prob(x<=-0.191919) <= 0.4
0.219862 <= Prob(x<=-0.171717) <= 0.4
0.221224 <= Prob(x<=-0.151515) <= 0.4
0.222474 <= Prob(x<=-0.131313) <= 0.4
0.223619 <= Prob(x<=-0.111111) <= 0.4
0.224669 <= Prob(x<=-0.0909091) <= 0.4
0.225643 <= Prob(x<=-0.0707071) <= 0.4
0.22653 <= Prob(x<=-0.0505051) <= 0.4
0.227334 <= Prob(x<=-0.030303) <= 0.4
0.3 <= Prob(x<=-0.010101) <= 0.4
0.3 <= Prob(x<=0.010101) <= 0.778942
0.3 <= Prob(x<=0.030303) <= 0.792532
0.3 <= Prob(x<=0.0505051) <= 0.806483
0.3 <= Prob(x<=0.0707071) <= 0.819022
0.3 <= Prob(x<=0.0909091) <= 0.825
0.3 <= Prob(x<=0.111111) <= 0.83125
0.3 <= Prob(x<=0.131313) <= 0.837791
0.3 <= Prob(x<=0.151515) <= 0.841937
0.3 <= Prob(x<=0.171717) <= 0.845957
0.3 <= Prob(x<=0.191919) <= 0.850137
0.3 <= Prob(x<=0.212121) <= 0.854492
0.3 <= Prob(x<=0.232323) <= 0.859052
0.3 <= Prob(x<=0.252525) <= 0.863811
0.3 <= Prob(x<=0.272727) <= 0.868817
0.3 <= Prob(x<=0.292929) <= 0.874066
0.3 <= Prob(x<=0.313131) <= 0.877055
0.3 <= Prob(x<=0.333333) <= 0.880067
0.3 <= Prob(x<=0.353535) <= 0.883272
0.300787 <= Prob(x<=0.373737) <= 0.886687
0.307695 <= Prob(x<=0.393939) <= 0.890333
0.314397 <= Prob(x<=0.414141) <= 0.894234
0.320909 <= Prob(x<=0.434343) <= 0.898418
0.327232 <= Prob(x<=0.454545) <= 0.902981
0.333379 <= Prob(x<=0.474747) <= 0.909013
0.339323 <= Prob(x<=0.494949) <= 0.916606
0.345134 <= Prob(x<=0.515152) <= 0.925292
0.350719 <= Prob(x<=0.535354) <= 0.935184
0.356201 <= Prob(x<=0.555556) <= 0.946304
0.361491 <= Prob(x<=0.575758) <= 0.958921
0.366603 <= Prob(x<=0.59596) <= 0.973265
0.371622 <= Prob(x<=0.616162) <= 0.989508
0.387329 <= Prob(x<=0.636364) <= 1
0.410495 <= Prob(x<=0.656566) <= 1
0.439031 <= Prob(x<=0.676768) <= 1
0.466372 <= Prob(x<=0.69697) <= 1
0.492663 <= Prob(x<=0.717172) <= 1
0.518025 <= Prob(x<=0.737374) <= 1
0.542592 <= Prob(x<=0.757576) <= 1
0.56651 <= Prob(x<=0.777778) <= 1
0.589941 <= Prob(x<=0.79798) <= 1
0.613125 <= Prob(x<=0.818182) <= 1
0.635881 <= Prob(x<=0.838384) <= 1
0.657609 <= Prob(x<=0.858586) <= 1
0.678314 <= Prob(x<=0.878788) <= 1
0.697846 <= Prob(x<=0.89899) <= 1
0.716238 <= Prob(x<=0.919192) <= 1
0.733536 <= Prob(x<=0.939394) <= 1
0.74974 <= Prob(x<=0.959596) <= 1
0.764914 <= Prob(x<=0.979798) <= 1
1 <= Prob(x<=1) <= 1