% Figure 7.4
% Boyd & Vandenberghe "Convex Optimization"
% Original version by Lieven Vandenberghe
% Updated for CVX by Michael Grant, 2005-12-19

% Generate the data
P = [0.70  0.10
     0.20  0.10
     0.05  0.70
     0.05  0.10];
[n,m] = size(P);

% Construct the tradeoff curve by finding the
% the Pareto optimal deterministic detectors,
% which are the curve's vertices

nopts   = 1000;
weights = logspace(-5,5,nopts);
obj     = [0;1];
inds    = ones(n,1);

% minimize  -t1'*q1 - w*t2'*q2
% s.t.      t1+t2 = 1,  t1,t2 \geq 0

next = 2;
for i = 1 : nopts,
   PW = P * diag( [ 1 ; weights(i) ] );
   [ maxvals, maxinds ] = max( PW' );  % max elt in each row
   if (~isequal(maxinds', inds(:,next-1)))
       inds(:,next) = maxinds';
       T = zeros(m,n);
       for j=1:n
          T(maxinds(1,j),j) = 1;
       end;
       obj(:,next) = 1-diag(T*P);
       next = next+1;
   end;
end;
plot(obj(1,:), obj(2,:),[0 1], [0 1],'--');
grid on
for i=2:size(obj,2)-1
   text(obj(1,i),obj(2,i),['a', num2str(i-1)]);
end;

% Minimax detector: not deterministic

cvx_begin
    variables T( m, n ) D( m, m )
    minimize max( D(1,2), D(2,1) )
    subject to
        D == T * P;
        sum( T, 1 ) == 1;
        T >= 0;
cvx_end

objmp = 1 - diag( D );
text( objmp(1), objmp(2), 'b' );
xlabel('P_{fp}'); ylabel('P_{fn}');

%print -deps roc.eps

 
Calling SDPT3 4.0: 10 variables, 5 equality constraints
------------------------------------------------------------

 num. of constraints =  5
 dim. of linear var  = 10
*******************************************************************
   SDPT3: Infeasible path-following algorithms
*******************************************************************
 version  predcorr  gam  expon  scale_data
    NT      1      0.000   1        0    
it pstep dstep pinfeas dinfeas  gap      prim-obj      dual-obj    cputime
-------------------------------------------------------------------
 0|0.000|0.000|1.3e+01|1.4e+01|1.0e+03| 2.000000e+01  0.000000e+00| 0:0:00| chol  1  1 
 1|1.000|1.000|4.3e-06|1.0e-01|4.6e+01| 9.382551e+00 -3.532711e+01| 0:0:00| chol  1  1 
 2|1.000|0.971|9.4e-07|1.3e-02|1.2e+00| 5.905323e-01 -5.324106e-01| 0:0:00| chol  1  1 
 3|0.963|1.000|4.9e-08|1.0e-03|2.5e-01| 2.572625e-01  1.055711e-02| 0:0:00| chol  1  1 
 4|1.000|0.922|7.7e-09|1.7e-04|3.8e-02| 1.821755e-01  1.443686e-01| 0:0:00| chol  1  1 
 5|0.985|0.934|2.6e-09|2.1e-05|1.9e-03| 1.671952e-01  1.653329e-01| 0:0:00| chol  1  1 
 6|0.988|0.988|2.3e-10|1.2e-06|2.4e-05| 1.666732e-01  1.666530e-01| 0:0:00| chol  1  1 
 7|0.989|0.989|1.2e-11|1.4e-08|2.6e-07| 1.666667e-01  1.666665e-01| 0:0:00| chol  1  1 
 8|0.999|0.994|6.4e-14|8.4e-11|4.1e-09| 1.666667e-01  1.666667e-01| 0:0:00|
  stop: max(relative gap, infeasibilities) < 1.49e-08
-------------------------------------------------------------------
 number of iterations   =  8
 primal objective value =  1.66666668e-01
 dual   objective value =  1.66666664e-01
 gap := trace(XZ)       = 4.08e-09
 relative gap           = 3.06e-09
 actual relative gap    = 2.88e-09
 rel. primal infeas (scaled problem)   = 6.45e-14
 rel. dual     "        "       "      = 8.41e-11
 rel. primal infeas (unscaled problem) = 0.00e+00
 rel. dual     "        "       "      = 0.00e+00
 norm(X), norm(y), norm(Z) = 1.9e+00, 3.5e-01, 8.9e-01
 norm(A), norm(b), norm(C) = 4.3e+00, 3.0e+00, 2.2e+00
 Total CPU time (secs)  = 0.08  
 CPU time per iteration = 0.01  
 termination code       =  0
 DIMACS: 9.7e-14  0.0e+00  9.4e-11  0.0e+00  2.9e-09  3.1e-09
-------------------------------------------------------------------
 
------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.166667