%0 Journal Article %T On Certain Conditions for Convex Optimization in Hilbert Spaces %J Khayyam Journal of Mathematics %I Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures) %Z 2423-4788 %A Okelo, Benard %D 2019 %\ 07/01/2019 %V 5 %N 2 %P 108-112 %! On Certain Conditions for Convex Optimization in Hilbert Spaces %K Optimization problem %K convex function %K Hilbert space %R 10.22034/kjm.2019.88084 %X 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 $f : \mathbb{R}^{n}\rightarrow \mathbb{R}$ and let $x\in \mathbb{R}^{n}$ be a local solution to the problem $\min_{x\in \mathbb{R}^{n}} f(x).$ Then $f'(x,d)\geq 0$ for every direction $d\in \mathbb{R}^{n}$ for which $f'(x,d)$ exists. Moreover, Let $f : \mathbb{R}^{n}\rightarrow \mathbb{R}$ be differentiable at $x^{*}\in \mathbb{R}^{n}.$ If $x^{*}$ is a local minimum of $f$, then $\nabla f(x^{*}) = 0.$ A simple application involving the Dirichlet problem is also given. %U https://www.kjm-math.org/article_88084_b5eebff35178eb5f92b22a462b6c4f8b.pdf