function val = max_adj_return_fn(wj, pj,qj)
% MAX_ADJ_RETURN_FN - Returns the mean divided by standard deviation of the horse track betting problem
%
% Note 1: wj has N-1 elments the last element wj(N) is computed from the previous N-1 i.e. = 1 - sum( wj )
%
% Note 2: we use a logistic regression approximation to map the inputs from (-\infty,+\infty) to the range (0,1)
%   
% Written by:
% -- 
% John L. Weatherwax                2006-12-31
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

% turn everything into columns: 
%
wj = wj(:); pj = pj(:); qj = qj(:); 

% compute the first N-1 logistic mapped log weights and then the Nth one from all the N-1 others: 
%
lwj = exp( wj )/( 1 + sum(exp(wj)) ); lwj = [ lwj; 1-sum(lwj) ];

% compute the mean
%
m = sum( pj .* lwj .* ( qj + 1 ) ) - 1;

% compute the  standard deviation: 
%
val = sqrt( sum( pj .* ( lwj .* ( qj + 1 ) - 1 - m ).^2 ) );

% return the (negative) ratio so we can use a minimization routine to find this maximum:
%
val = -m / val;