Fractional Gradient Methods via ψ-Hilfer Derivative

Abstract

Motivated by the increase in practical applications of fractional calculus, we study the classical gradient method under the perspective of the ψ-Hilfer derivative. This allows us to cover several definitions of fractional derivatives that are found in the literature in our study. The convergence of the ψ-Hilfer continuous fractional gradient method was studied both for strongly and non-strongly convex cases. Using a series representation of the target function, we developed an algorithm for the ψ-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 step size optimization, the ψ-Hilfer fractional gradient method showed better results in terms of speed and accuracy. Our results generalize previous works in the literature.