On 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  $f : \mathbb{R}^{n}\rightarrow \mathbb{R}$ 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. Lastly, we have given optimization conditions involving positive semi-definite matrices.

Keywords

• Optimization,Convexity,Hilbert space

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