{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T05:51:06Z","timestamp":1760248266306,"version":"build-2065373602"},"reference-count":22,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2023,3,21]],"date-time":"2023-03-21T00:00:00Z","timestamp":1679356800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CIDMA (Center for Research and Development in Mathematics and Applications)","award":["UIDB\/04106\/2020","UIDP\/04106\/2020","CEECIND\/01131\/2018"],"award-info":[{"award-number":["UIDB\/04106\/2020","UIDP\/04106\/2020","CEECIND\/01131\/2018"]}]},{"name":"FCT (Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia)","award":["UIDB\/04106\/2020","UIDP\/04106\/2020","CEECIND\/01131\/2018"],"award-info":[{"award-number":["UIDB\/04106\/2020","UIDP\/04106\/2020","CEECIND\/01131\/2018"]}]},{"name":"FCT via the 2018 FCT program of Stimulus of Scientific Employment\u2014Individual Support","award":["UIDB\/04106\/2020","UIDP\/04106\/2020","CEECIND\/01131\/2018"],"award-info":[{"award-number":["UIDB\/04106\/2020","UIDP\/04106\/2020","CEECIND\/01131\/2018"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>Motivated by the increase in practical applications of fractional calculus, we study the classical gradient method under the perspective of the \u03c8-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 \u03c8-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 \u03c8-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 \u03c8-Hilfer fractional gradient method showed better results in terms of speed and accuracy. Our results generalize previous works in the literature.<\/jats:p>","DOI":"10.3390\/fractalfract7030275","type":"journal-article","created":{"date-parts":[[2023,3,21]],"date-time":"2023-03-21T06:56:48Z","timestamp":1679381808000},"page":"275","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Fractional Gradient Methods via \u03c8-Hilfer Derivative"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8756-4893","authenticated-orcid":false,"given":"Nelson","family":"Vieira","sequence":"first","affiliation":[{"name":"CIDMA \u2014 Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universit\u00e1rio de Santiago, 3810-193 Aveiro, Portugal"}]},{"given":"M. Manuela","family":"Rodrigues","sequence":"additional","affiliation":[{"name":"CIDMA \u2014 Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus Universit\u00e1rio de Santiago, 3810-193 Aveiro, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1816-8293","authenticated-orcid":false,"given":"Milton","family":"Ferreira","sequence":"additional","affiliation":[{"name":"School of Technology and Management, Polytechnic of Leiria, 2411-901 Leiria, Portugal"},{"name":"CIDMA\u2014Center for Research and Development in Mathematics and Applications, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,21]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2916","DOI":"10.1016\/j.automatica.2008.04.008","article-title":"New IIR filter-based adaptive algorithm in active noise control applications: Commutation error-introduced LMS algorithm and associated convergence assessment by a deterministic approach","volume":"44","author":"Lin","year":"2008","journal-title":"Automatica"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1109\/TNNLS.2013.2286175","article-title":"Fractional extreme value adaptive training method: Fractional steepest descent approach","volume":"26","author":"Pu","year":"2015","journal-title":"IEEE Trans. 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