{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T03:35:16Z","timestamp":1777088116675,"version":"3.51.4"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2025,12,25]],"date-time":"2025-12-25T00:00:00Z","timestamp":1766620800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>An upgrade to the quasi-Newton (QN) family of methods for solving unconstrained optimization problems is proposed. This research focuses on a detailed investigation of the Barzilai and Borwein (BB) gradient methods. The upgrade involves the use of neutrosophic logic to determine an additional parameter that will be incorporated into an appropriate step size for the BB iterations. Unlike previous research, which incorporated neutrosophic concepts into gradient methods by using only two objective-function values to calculate the input parameter during the neutrophication phase, this study determines the input parameter using three consecutive objective-function values. The main idea is to use appropriately defined membership functions to perform neutrosophication and de-neutrosophication. The set of if\u2013then rules is based on two or more successive values of the objective function. This strategy also directly influences the design of the newly proposed method. Numerical comparisons demonstrate superior performance of the proposed methods with respect to Dolan\u2013Mor\u00e9 performance profiles including the number of iterations, central processing unit (CPU) time, and number of function evaluations. Furthermore, experimental results confirm that the proposed algorithms can be effectively applied to image restoration tasks, particularly for image denoising, where they achieve competitive reconstruction quality and stable convergence behavior.<\/jats:p>","DOI":"10.3390\/axioms15010011","type":"journal-article","created":{"date-parts":[[2025,12,25]],"date-time":"2025-12-25T23:52:27Z","timestamp":1766706747000},"page":"11","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Improvement of Barzilai and Borwein Gradient Method Based on Neutrosophic Logic System with Application in Image Restoration"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0655-3741","authenticated-orcid":false,"given":"Predrag S.","family":"Stanimirovi\u0107","sequence":"first","affiliation":[{"name":"Faculty of Sciences and Mathematics, University of Ni\u0161, Vi\u0161egradska 33, 18000 Ni\u0161, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9179-0965","authenticated-orcid":false,"given":"Branislav D.","family":"Ivanov","sequence":"additional","affiliation":[{"name":"Faculty of Sciences and Mathematics, University of Ni\u0161, Vi\u0161egradska 33, 18000 Ni\u0161, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0494-3201","authenticated-orcid":false,"given":"Marko","family":"Miladinovi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Sciences and Mathematics, University of Ni\u0161, Vi\u0161egradska 33, 18000 Ni\u0161, Serbia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6846-3074","authenticated-orcid":false,"given":"Dragi\u0161a","family":"Stanujki\u0107","sequence":"additional","affiliation":[{"name":"Technical Faculty in Bor, University of Belgrade, Vojske Jugoslavije 12, 19210 Bor, Serbia"},{"name":"College of Global Business, Korea University, Sejong 30019, Republic of Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Nocedal, J., and Wright, J. 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