Fractional gradient methods via ψ-Hilfer derivative

Abstract

Motivated by the increasing of practical applications in fractional calculus, we study the classical gradient method under the perspective of the $\psi$-Hilfer derivative. This allows us to cover in our study several definitions of fractional derivatives that are found in the literature. The convergence of the $\psi$-Hilfer continuous fractional gradient method is studied both for strongly and non-strongly convex cases. Using a series representation of the target function, we develop an algorithm for the $\psi$-Hilfer fractional order gradient method. The numerical method obtained by truncating higher-order terms was tested and analyzed using benchmark functions. Considering variable order differentiation and optimizing the step size, the $\psi$-Hilfer fractional gradient method shows better results in terms of speed and accuracy. Our results generalize previous works in the literature.