  This paper introduces a new functional optimization approach to portfolio
optimization problems by treating the unknown weight vector as a function of
past values instead of treating them as fixed unknown coefficients in the
majority of studies. We first show that the optimal solution, in general, is
not a constant function. We give the optimal conditions for a vector function
to be the solution, and hence give the conditions for a plug-in solution
(replacing the unknown mean and variance by certain estimates based on past
values) to be optimal. After showing that the plug-in solutions are sub-optimal
in general, we propose gradient-ascent algorithms to solve the functional
optimization for mean-variance portfolio management with theorems for
convergence provided. Simulations and empirical studies show that our approach
can perform significantly better than the plug-in approach.
