P = [0.70 0.10
0.20 0.10
0.05 0.70
0.05 0.10];
[n,m] = size(P);
nopts = 1000;
weights = logspace(-5,5,nopts);
obj = [0;1];
inds = ones(n,1);
next = 2;
for i = 1 : nopts,
PW = P * diag( [ 1 ; weights(i) ] );
[ maxvals, maxinds ] = max( PW' );
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;
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}');
Calling Mosek 9.1.9: 10 variables, 5 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 : 5
Cones : 0
Scalar variables : 10
Matrix variables : 0
Integer variables : 0
Optimizer started.
Presolve started.
Linear dependency checker started.
Linear dependency checker terminated.
Eliminator started.
Freed constraints in eliminator : 4
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 : 5
Cones : 0
Scalar variables : 10
Matrix variables : 0
Integer variables : 0
Optimizer - threads : 8
Optimizer - solved problem : the primal
Optimizer - Constraints : 1
Optimizer - Cones : 0
Optimizer - Scalar variables : 6 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 : 1 after factor : 1
Factor - dense dim. : 0 flops : 1.30e+01
ITE PFEAS DFEAS GFEAS PRSTATUS POBJ DOBJ MU TIME
0 2.0e+00 2.0e+00 6.0e+00 0.00e+00 2.000000000e+00 0.000000000e+00 4.0e+00 0.00
1 4.1e-01 4.1e-01 1.2e+00 2.33e+00 5.271701753e-01 4.421583667e-01 8.1e-01 0.01
2 4.8e-02 4.8e-02 1.4e-01 1.01e+00 2.187626321e-01 2.057396942e-01 9.6e-02 0.01
3 5.4e-03 5.4e-03 1.6e-02 9.95e-01 1.722986032e-01 1.704628420e-01 1.1e-02 0.01
4 6.9e-05 6.9e-05 2.1e-04 1.00e+00 1.666975808e-01 1.666681082e-01 1.4e-04 0.01
5 7.0e-09 7.0e-09 2.1e-08 1.00e+00 1.666666698e-01 1.666666668e-01 1.4e-08 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: 0.01
Interior-point solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 1.6666666981e-01 nrm: 1e+00 Viol. con: 2e-09 var: 2e-09
Dual. obj: 1.6666666684e-01 nrm: 7e-01 Viol. con: 0e+00 var: 2e-09
Basic solution summary
Problem status : PRIMAL_AND_DUAL_FEASIBLE
Solution status : OPTIMAL
Primal. obj: 1.6666666667e-01 nrm: 1e+00 Viol. con: 0e+00 var: 3e-17
Dual. obj: 1.6666666251e-01 nrm: 7e-01 Viol. con: 0e+00 var: 7e-18
Optimizer summary
Optimizer - time: 0.01
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): +0.166667
