{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,9]],"date-time":"2026-05-09T02:09:03Z","timestamp":1778292543691,"version":"3.51.4"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T00:00:00Z","timestamp":1757462400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T00:00:00Z","timestamp":1757462400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Optim Theory Appl"],"published-print":{"date-parts":[[2026,1]]},"DOI":"10.1007\/s10957-025-02826-x","type":"journal-article","created":{"date-parts":[[2025,9,10]],"date-time":"2025-09-10T18:46:53Z","timestamp":1757530013000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["A Unified Derivative-Free Projection Framework for Convex-Constrained Nonlinear Equations"],"prefix":"10.1007","volume":"208","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5534-1759","authenticated-orcid":false,"given":"Abdulkarim Hassan","family":"Ibrahim","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1367-646X","authenticated-orcid":false,"given":"Mohammed","family":"Alshahrani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5998-0635","authenticated-orcid":false,"given":"Suliman","family":"Al-Homidan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,10]]},"reference":[{"key":"2826_CR1","doi-asserted-by":"publisher","first-page":"92157","DOI":"10.1109\/ACCESS.2021.3091906","volume":"9","author":"AB Abubakar","year":"2021","unstructured":"Abubakar, A.B., Kumam, P., Ibrahim, A.H.: Inertial derivative-free projection method for nonlinear monotone operator equations with convex constraints. IEEE Access 9, 92157\u201392167 (2021)","journal-title":"IEEE Access"},{"issue":"1","key":"2826_CR2","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1023\/A:1011253113155","volume":"9","author":"F Alvarez","year":"2001","unstructured":"Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal. 9(1), 3\u201311 (2001)","journal-title":"Set-Valued Anal."},{"issue":"1","key":"2826_CR3","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1137\/141000671","volume":"59","author":"J Bezanson","year":"2017","unstructured":"Bezanson, J., Edelman, A., Karpinski, S., Shah, V.B.: Julia: A fresh approach to numerical computing. SIAM Rev. 59(1), 65\u201398 (2017). https:\/\/doi.org\/10.1137\/141000671","journal-title":"SIAM Rev."},{"key":"2826_CR4","doi-asserted-by":"publisher","unstructured":"Burdakov, O., Dai, Y., Huang, N.: Stabilized Barzilai-Borwein method. J. Comput. Math. 37(6), 916\u2013936 (2019). https:\/\/doi.org\/10.4208\/jcm.1911-m2019-0171","DOI":"10.4208\/jcm.1911-m2019-0171"},{"issue":"4","key":"2826_CR5","doi-asserted-by":"publisher","first-page":"319","DOI":"10.1080\/10556789508805619","volume":"5","author":"SP Dirkse","year":"1995","unstructured":"Dirkse, S.P., Ferris, M.C.: Mcplib: A collection of nonlinear mixed complementarity problems. Optim. Methods Softw. 5(4), 319\u2013345 (1995)","journal-title":"Optim. Methods Softw."},{"issue":"2","key":"2826_CR6","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1007\/s101070100263","volume":"91","author":"ED Dolan","year":"2002","unstructured":"Dolan, E.D., Mor\u00e9, J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2), 201\u2013213 (2002)","journal-title":"Math. Program."},{"issue":"4","key":"2826_CR7","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1007\/s10092-018-0291-2","volume":"55","author":"P Gao","year":"2018","unstructured":"Gao, P., He, C.: An efficient three-term conjugate gradient method for nonlinear monotone equations with convex constraints. Calcolo 55(4), 53 (2018). https:\/\/doi.org\/10.1007\/s10092-018-0291-2","journal-title":"Calcolo"},{"issue":"7","key":"2826_CR8","doi-asserted-by":"publisher","first-page":"306","DOI":"10.1007\/s40314-023-02446-z","volume":"42","author":"ML Gon\u00e7alves","year":"2023","unstructured":"Gon\u00e7alves, M.L., Menezes, T.C.: A framework for convex-constrained monotone nonlinear equations and its special cases. Comput. Appl. Math. 42(7), 306 (2023)","journal-title":"Comput. Appl. Math."},{"issue":"4","key":"2826_CR9","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1145\/962437.962439","volume":"29","author":"NI Gould","year":"2003","unstructured":"Gould, N.I., Orban, D., Toint, P.L.: Cuter and sifdec: A constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29(4), 373\u2013394 (2003). https:\/\/doi.org\/10.1145\/962437.962439","journal-title":"ACM Trans. Math. Softw."},{"issue":"116","key":"2826_CR10","first-page":"071","volume":"451","author":"AH Ibrahim","year":"2024","unstructured":"Ibrahim, A.H., Al-Homidan, S.: Two-step inertial derivative-free projection method for solving nonlinear equations with application. J. Comput. Appl. Math. 451(116), 071 (2024)","journal-title":"J. Comput. Appl. Math."},{"key":"2826_CR11","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1016\/j.apnum.2024.11.006","volume":"209","author":"AH Ibrahim","year":"2025","unstructured":"Ibrahim, A.H., Al-Homidan, S.: A derivative-free projection method with double inertial effects for solving nonlinear equations. Appl. Numer. Math. 209, 55\u201367 (2025)","journal-title":"Appl. Numer. Math."},{"issue":"4","key":"2826_CR12","doi-asserted-by":"publisher","first-page":"1625","DOI":"10.1007\/s11075-023-01679-7","volume":"96","author":"AH Ibrahim","year":"2024","unstructured":"Ibrahim, A.H., Alshahrani, M., Al-Homidan, S.: Two classes of spectral three-term derivative-free method for solving nonlinear equations with application. Numer. Algorithms 96(4), 1625\u20131645 (2024)","journal-title":"Numer. Algorithms"},{"key":"2826_CR13","doi-asserted-by":"crossref","unstructured":"Ibrahim, A.H., Alshahrani, M., Al-Homidan, S.: A class of relaxed-inertial derivative-free projection method beyond monotonicity with application. Arab. J. Math. pp. 1\u201332 (2025)","DOI":"10.1007\/s40065-024-00491-y"},{"issue":"1","key":"2826_CR14","doi-asserted-by":"publisher","first-page":"30","DOI":"10.3934\/jimo.2021173","volume":"19","author":"AH Ibrahim","year":"2022","unstructured":"Ibrahim, A.H., Kumam, P., Sun, M., Chaipunya, P., Abubakar, A.B.: Projection method with inertial step for nonlinear equations: application to signal recovery. J. Ind. Manag. Optim. 19(1), 30\u201355 (2022)","journal-title":"J. Ind. Manag. Optim."},{"issue":"7","key":"2826_CR15","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s40314-022-02019-6","volume":"41","author":"J Jian","year":"2022","unstructured":"Jian, J., Yin, J., Tang, C., Han, D.: A family of inertial derivative-free projection methods for constrained nonlinear pseudo-monotone equations with applications. Comput. Appl. Math. 41(7), 309 (2022)","journal-title":"Comput. Appl. Math."},{"issue":"4","key":"2826_CR16","doi-asserted-by":"publisher","first-page":"445","DOI":"10.1007\/BF00932654","volume":"18","author":"S Karamardian","year":"1976","unstructured":"Karamardian, S.: Complementarity problems over cones with monotone and pseudomonotone maps. J. Optim. Theory Appl. 18(4), 445\u2013454 (1976). https:\/\/doi.org\/10.1007\/BF00932654","journal-title":"J. Optim. Theory Appl."},{"key":"2826_CR17","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2025.116546","volume":"464","author":"X Ma","year":"2025","unstructured":"Ma, X., Yang, S., Liu, P.: Two inertial-relaxed hybridized cg projection-based algorithms for solving nonlinear monotone equations applied in image restoration. J. Comput. Appl. Math. 464, 116546 (2025)","journal-title":"J. Comput. Appl. Math."},{"issue":"4","key":"2826_CR18","first-page":"333","volume":"22","author":"K Meintjes","year":"1987","unstructured":"Meintjes, K., Morgan, A.P.: A methodology for solving chemical equilibrium systems. Appl. Math. Comput. 22(4), 333\u2013361 (1987)","journal-title":"Appl. Math. Comput."},{"issue":"4","key":"2826_CR19","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1007\/s11075-022-01483-9","volume":"93","author":"Y Ou","year":"2023","unstructured":"Ou, Y., Li, L.: A unified convergence analysis of the derivative-free projection-based method for constrained nonlinear monotone equations. Numer. Algorithms 93(4), 1639\u20131660 (2023)","journal-title":"Numer. Algorithms"},{"key":"2826_CR20","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1016\/j.apnum.2022.10.016","volume":"184","author":"J Sabi\u2019u","year":"2023","unstructured":"Sabi\u2019u, J., Shah, A., Stanimirovi\u0107, P.S., Ivanov, B., Waziri, M.Y.: Modified optimal perry conjugate gradient method for solving system of monotone equations with applications. Appl. Numer. Math. 184, 431\u2013445 (2023). https:\/\/doi.org\/10.1016\/j.apnum.2022.10.016. (https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0168927422002860)","journal-title":"Appl. Numer. Math."},{"issue":"1","key":"2826_CR21","doi-asserted-by":"publisher","first-page":"17","DOI":"10.5614\/j.math.fund.sci.2020.52.1.2","volume":"52","author":"J Sabi\u2019u","year":"2020","unstructured":"Sabi\u2019u, J., Shah, A., Waziri, M.Y., Dauda, M.K.: A new hybrid approach for solving large-scale monotone nonlinear equations. J. Math. Fundam. Sci. 52(1), 17\u201326 (2020). https:\/\/doi.org\/10.5614\/j.math.fund.sci.2020.52.1.2","journal-title":"J. Math. Fundam. Sci."},{"issue":"3","key":"2826_CR22","doi-asserted-by":"publisher","first-page":"877","DOI":"10.1007\/s10957-019-01616-6","volume":"184","author":"Y Shehu","year":"2020","unstructured":"Shehu, Y., Gibali, A., Sagratella, S.: Inertial projection-type methods for solving quasi-variational inequalities in real hilbert spaces. J. Optim. Theory Appl. 184(3), 877\u2013894 (2020)","journal-title":"J. Optim. Theory Appl."},{"key":"2826_CR23","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1007\/978-1-4757-6388-1_18","volume-title":"A Globally Convergent Inexact Newton Method for Systems of Monotone Equations","author":"MV Solodov","year":"1999","unstructured":"Solodov, M.V., Svaiter, B.F.: A Globally Convergent Inexact Newton Method for Systems of Monotone Equations, pp. 355\u2013369. Springer, US, Boston, MA (1999). https:\/\/doi.org\/10.1007\/978-1-4757-6388-1_18"},{"issue":"115","key":"2826_CR24","first-page":"873","volume":"446","author":"T Song","year":"2024","unstructured":"Song, T., Liu, Z.: An efficient inertial subspace minimization CG algorithm with convergence rate analysis for constrained nonlinear monotone equations. J. Comput. Appl. Math. 446(115), 873 (2024)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"2826_CR25","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s00186-006-0140-y","volume":"66","author":"C Wang","year":"2007","unstructured":"Wang, C., Wang, Y., Xu, C.: A projection method for a system of nonlinear monotone equations with convex constraints. Math. Methods Oper. Res. 66(1), 33\u201346 (2007). https:\/\/doi.org\/10.1007\/s00186-006-0140-y","journal-title":"Math. Methods Oper. Res."},{"key":"2826_CR26","doi-asserted-by":"crossref","unstructured":"Wood, A., Wollenberg, B.: Power Generation Operation and Control. John Wiley & Sons Inc, New York (1996)","DOI":"10.1016\/0140-6701(96)88715-7"},{"issue":"10","key":"2826_CR27","doi-asserted-by":"publisher","first-page":"2416","DOI":"10.1016\/j.apnum.2009.04.004","volume":"59","author":"Z Yu","year":"2009","unstructured":"Yu, Z., Lin, J., Sun, J., Xiao, Y., Liu, L., Li, Z.: Spectral gradient projection method for monotone nonlinear equations with convex constraints. Appl. Numer. Math. 59(10), 2416\u20132423 (2009). https:\/\/doi.org\/10.1016\/j.apnum.2009.04.004. (https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0168927409000580)","journal-title":"Appl. Numer. Math."},{"key":"2826_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.112781","volume":"375","author":"L Zheng","year":"2020","unstructured":"Zheng, L., Yang, L., Liang, Y.: A conjugate gradient projection method for solving equations with convex constraints. J. Comput. Appl. Math. 375, 112781 (2020). https:\/\/doi.org\/10.1016\/j.cam.2020.112781. (https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0377042720300728)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"2826_CR29","first-page":"89","volume":"25","author":"W Zhou","year":"2007","unstructured":"Zhou, W., Li, D.: Limited memory BFGS method for nonlinear monotone equations. J. Comput. Math. 25(1), 89\u201396 (2007)","journal-title":"J. Comput. Math."}],"container-title":["Journal of Optimization Theory and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02826-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10957-025-02826-x","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10957-025-02826-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T07:22:28Z","timestamp":1775028148000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10957-025-02826-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,10]]},"references-count":29,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2026,1]]}},"alternative-id":["2826"],"URL":"https:\/\/doi.org\/10.1007\/s10957-025-02826-x","relation":{},"ISSN":["0022-3239","1573-2878"],"issn-type":[{"value":"0022-3239","type":"print"},{"value":"1573-2878","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,10]]},"assertion":[{"value":"13 February 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 August 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 September 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflicts of Interest"}},{"value":"This article does not contain any studies with human participants or animals performed by any of the authors.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical Approval"}},{"order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Code availability"}}],"article-number":"11"}}