ON LIPSCHITZ FUNCTIONS AND OPTIMIZATION IN HILBERT SPACES

LIPSCHITZ FUNCTIONS AND OPTIMIZATION

Abstract

Optimization problems have attracted the attention of many mathematicians and researchers over a long period of time. In this work, we consider with the classical results on optimization of functions in infinite-dimensional real Hilbert spaces. In particular, we consider Lipschitz functions. The methodology involves the use of convex optimization techniques. The results show that a Lipschitz function f has a first order optimality condition. Moreover, if f is differentiable at a point x* in Rⁿ and if x* is a local minimum of f, then the del of f(x*) = 0. A simple application involving the Dirichlet problem is also given.  In conclusion, we give the applications of this work to Portfolio Optimization, particularly, stochastic optimization with consideration to Hamilton-Jacobi-Bellman Equation in Hilbert spaces.