On Certain Conditions for Convex Optimization in Hilbert Spaces

Abstract

In this paper convex optimization techniques are employed for convex optimization problems in infinite dimensional Hilbert spaces. A first order optimality condition is given. Let $f:{\mathbb{R}}^n\to \mathbb{R}$ and let $x\in {\mathbb{R}}^n$ be a local solution to the problem ${\min }_{x\in {\mathbb{R}}^n}f(x).$ Then $f^{\prime }(x,d)\ge 0$ for every direction $d\in {\mathbb{R}}^n$ for which $f^{\prime }(x,d)$ exists. Moreover, Let $f:{\mathbb{R}}^n\to \mathbb{R}$ be differentiable at $x^{\ast }\in {\mathbb{R}}^n.$ If $x^{\ast }$ is a local minimum of $f$, then $\nabla f(x^{\ast })=0.$ A simple application involving the Dirichlet problem is also given.