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.