Restricted multinomial maximum likelihood estimation based upon Fenchel duality

A commonly occurring problem is that of maximizing a multinomial likelihood over a restricted region. We show that if the region is convex, then a dual problem always exists which is frequently more tractable. A solution to the dual problem leads directly to a solution for the original problem and conversely. Moreover, the form of the dual problem suggests an iterative algorithm for solving a MLE problem when the constraint region can be written as a finite intersection of 'nice' constraint regions. We show that this iterative algorithm is guaranteed to converge to the true solution and give several meaningful examples of the algorithm.