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Title: Fractional gradient methods via ψ-Hilfer derivative
Author: Vieira, N.
Rodrigues, M. M.
Ferreira, M.
Keywords: Fractional calculus
$\psi$-Hilfer fractional derivative
Fractional gradient method
Issue Date: Mar-2023
Publisher: MDPI
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.
Peer review: yes
DOI: 10.3390/fractalfract7030275
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Appears in Collections:CIDMA - Artigos
CHAG - Artigos

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