Example 7.4: Binary hypothesis testing
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 SeDuMi: 11 variables (1 free), 6 equality constraints
------------------------------------------------------------------------
SeDuMi 1.1 by AdvOL, 2005 and Jos F. Sturm, 1998, 2001-2003.
Alg = 2: xz-corrector, Adaptive Step-Differentiation, theta = 0.250, beta = 0.500
Split 1 free variables
eqs m = 6, order n = 13, dim = 13, blocks = 1
nnz(A) = 22 + 0, nnz(ADA) = 24, nnz(L) = 15
it : b*y gap delta rate t/tP* t/tD* feas cg cg prec
0 : 2.33E+000 0.000
1 : 3.77E-001 5.63E-001 0.000 0.2420 0.9000 0.9000 3.01 1 1 6.9E-001
2 : 2.04E-001 1.17E-001 0.000 0.2070 0.9000 0.9000 1.27 1 1 1.3E-001
3 : 1.77E-001 2.70E-002 0.000 0.2315 0.9000 0.9000 1.05 1 1 3.0E-002
4 : 1.68E-001 2.25E-003 0.000 0.0835 0.9900 0.9900 1.01 1 1 2.5E-003
5 : 1.67E-001 8.62E-007 0.000 0.0004 0.9999 0.9999 1.00 1 1
iter seconds digits c*x b*y
5 0.1 15.5 1.6666666667e-001 1.6666666667e-001
|Ax-b| = 3.6e-016, [Ay-c]_+ = 1.6E-016, |x|= 2.2e+000, |y|= 8.7e-001
Detailed timing (sec)
Pre IPM Post
1.001E-002 7.010E-002 0.000E+000
Max-norms: ||b||=1, ||c|| = 1,
Cholesky |add|=0, |skip| = 0, ||L.L|| = 1.
------------------------------------------------------------------------
Status: Solved
Optimal value (cvx_optval): +0.166667