On Convex Optimization in Hilbert Spaces

Authors : Benard OKELO

Pages : 89-95

View : 22 | Download : 15

Publication Date : 2019-10-30

Article Type : Research Paper

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  $f : \mathbb{R}^{n}\rightarrow \mathbb{R}$ and let $x\in \mathbb{R}^{n}$ be a local solution to the problem $\min_{x\in \mathbb{R}^{n}} finsert ignore into journalissuearticles values(x);.$ Then $f`insert ignore into journalissuearticles values(x,d);\geq 0$ for every direction $d\in \mathbb{R}^{n}$  for which $f`insert ignore into journalissuearticles values(x,d);$ 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 $\nabla finsert ignore into journalissuearticles values(x^{*}); = 0.$ A simple application involving the Dirichlet problem is also given. Lastly, we have given optimization conditions involving positive semi-definite matrices.
Keywords : Optimization, Convexity, Hilbert space