To solve either (P) or (R), we take as our guiding philosophy:
do not discretize until (unless) necessary. In many cases, we
can actually characterize solutions to (P) and (R) by solving a
system of nonlinear equations in 
, which we can then solve
numerically.
In short, our characterization is obtained by considering the dual problems:
and
where 
can be explicitly calculated, and 
.
For example, in maximum entropy problems, we begin with
with dual problem
To solve 
, we observe that 
is concave, and so a maximum occurs exactly
when 
. That is, we solve 
for 
, which is a system of nonlinear equations:
Once the optimal dual vector 
is obtained, we can recover
the optimal function 
from the formula
when 
.
In the ME case, 
, 
, so
the maximum entropy solution is given by
which is a functional form of the solution.