Frechet Discrete Gradient and Hessian Operators on Infinite-Dimensional Spaces

Abstract

Benefiting from the notion of Frechet derivatives, we define Frechet discrete operators, such as gradient and Hessian, on infinite-dimensional spaces. The Frechet discrete gradient expands upon the concept of the discrete gradient of Gonzalez (1996) for finite-dimensional spaces. The Frechet discrete Hessian elevates the property to second-order representations of the Frechet derivative. By leveraging these operators, we offer an initial exploration of discrete gradient methods for convex optimization in infinite-dimensional spaces. Under mild conditions on the objective functional, we establish the convergence of any sequence generated by the proposed Frechet discrete gradient method, regardless of the choice of the finite learning rate. Copyright (C) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)