Fréchet Discrete Gradient and Hessian Operators on Infinite-Dimensional Spaces

Benefiting from the notion of Fréchet derivatives, we define Fréchet discrete operators, such as gradient and Hessian, on infinite-dimensional spaces. The Fréchet discrete gradient expands upon the concept of the discrete gradient of Gonzalez (1996) for finite-dimensional spaces. The Fréchet discrete Hessian elevates the property to second-order representations of the Fréchet 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 Fréchet discrete gradient method, regardless of the choice of the finite learning rate.