On Certain Conditions for Convex Optimization in Hilbert Spaces

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}\rightarrow $\mathbb{R}$$andlet$x\in $\mathbb{R}$^{n}$bealocalsolutiontotheproblem$\min_{x\in $\mathbb{R}$^{n}} f(x).$Then$f’(x,d)\geq 0$foreverydirection$d\in $\mathbb{R}$^{n}$forwhich$f’(x,d)$exists\mathrm{.}Moreover,Let$f : $\mathbb{R}$^{n}\rightarrow $\mathbb{R}$$bedifferentiableat$x^{}\in $\mathbb{R}$^{n}.$If$x^{}$isalocalminimumof$f$,then$\nabla f(x^{*}) = 0.$ A simple application involving the Dirichlet problem is also given.