A modified version of RMIL conjugate gradient method and its application with backpropagation in machine learning

A modified version of RMIL conjugate gradient method and its application with backpropagation in machine learning | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article A modified version of RMIL conjugate gradient method and its application with backpropagation in machine learning Osman Omer Osman Yousif, Yasser Nourain, Mohammed A. Saleh, Abdulgader Z. Almaymuni This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9086018/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract The computational efficiency and low memory requirements make the conjugate gradient (CG) attractive methods for solving large-scale unconstrained optimization problems. In 2012, a CG method called RMIL was proposed. Later, in 2016, it was modified to RMIL+, which proved to be convergent under the exact and the strong Wolfe line searches. In 2024, the RMIL + was improved to RMIL* method. In this paper, based on the construction of RMIL* method, we propose another modified version of RMIL. Under mild conditions, we present the proof of this new method. Moreover, a numerical experiment based on some comparisons with other CG methods was conducted. Additionally, the new method was applied in machine learning to train some neural network models. The results demonstrate the validity of the proposed method. Mathematics Subject Classification : 65K05, 90C30, 90C56 optimization method conjugate gradient methods RMIL conjugate gradient method global convergence machine learning Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 19 Apr, 2026 Reviews received at journal 25 Mar, 2026 Reviewers agreed at journal 18 Mar, 2026 Reviewers invited by journal 17 Mar, 2026 Editor assigned by journal 11 Mar, 2026 Submission checks completed at journal 10 Mar, 2026 First submitted to journal 10 Mar, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9086018","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":608430573,"identity":"8a9d16c4-4012-4b06-8601-7750653aeb60","order_by":0,"name":"Osman Omer Osman Yousif","email":"","orcid":"","institution":"University of Gezira","correspondingAuthor":false,"prefix":"","firstName":"Osman","middleName":"Omer Osman","lastName":"Yousif","suffix":""},{"id":608430574,"identity":"f4f88c1a-9345-4894-b6f7-737d90d6f53f","order_by":1,"name":"Yasser Nourain","email":"","orcid":"","institution":"University of Gezira","correspondingAuthor":false,"prefix":"","firstName":"Yasser","middleName":"","lastName":"Nourain","suffix":""},{"id":608430575,"identity":"45a6976d-c78a-4c36-b17b-26bc73371bd6","order_by":2,"name":"Mohammed A. Saleh","email":"","orcid":"","institution":"Qassim University","correspondingAuthor":false,"prefix":"","firstName":"Mohammed","middleName":"A.","lastName":"Saleh","suffix":""},{"id":608430576,"identity":"eb8f2755-4e70-4279-9a57-1693d8e15db0","order_by":3,"name":"Abdulgader Z. Almaymuni","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVRIiWNgGAWjYDACdijNxt4AopiJ0AJTw8ZzgFQtDBIJRGrhb+ZO/MybcziPT/LxNgmGCuvEBv7DD/BqkTjMu1mad9vhYjbptDIJhjPpiQ0SaQb4rTnMuwGkJbFNOsdMgrHtMFALA34t8kBbfoO1SJ4BavkH1MJ//ANeLQaHebdBbJHgAWppAGphyMFviyFQi+XcbemJbTxpxRYJx9KN2yRyCvBqkTveu/nG223WifPbD2+88aHGWraf//gGvFpAgIkH6kiGBCDJRlA9EDD+gGkZBaNgFIyCUYANAADthELQaV9W5QAAAABJRU5ErkJggg==","orcid":"","institution":"Qassim University","correspondingAuthor":true,"prefix":"","firstName":"Abdulgader","middleName":"Z.","lastName":"Almaymuni","suffix":""}],"badges":[],"createdAt":"2026-03-10 15:54:25","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9086018/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9086018/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105035841,"identity":"d971a723-e35f-4be8-b940-947e52e28a8e","added_by":"auto","created_at":"2026-03-20 07:26:44","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":596864,"visible":true,"origin":"","legend":"","description":"","filename":"AmodifiedRMILPlus1.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9086018/v1_covered_ec91c8ae-64e2-4deb-bdcf-42f13bb10a55.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A modified version of RMIL conjugate gradient method and its application with backpropagation in machine learning","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-inequalities-and-applications","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jiap","sideBox":"Learn more about [Journal of Inequalities and Applications](http://journalofinequalitiesandapplications.springeropen.com)","snPcode":"13660","submissionUrl":"https://submission.nature.com/new-submission/13660/3","title":"Journal of Inequalities and Applications","twitterHandle":"@SpringerMath","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Open","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"optimization method, conjugate gradient methods, RMIL conjugate gradient method, global convergence, machine learning","lastPublishedDoi":"10.21203/rs.3.rs-9086018/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9086018/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe computational efficiency and low memory requirements make the conjugate gradient (CG) attractive methods for solving large-scale unconstrained optimization problems. In 2012, a CG method called RMIL was proposed. Later, in 2016, it was modified to RMIL+, which proved to be convergent under the exact and the strong Wolfe line searches. In 2024, the RMIL + was improved to RMIL* method. In this paper, based on the construction of RMIL* method, we propose another modified version of RMIL. Under mild conditions, we present the proof of this new method. Moreover, a numerical experiment based on some comparisons with other CG methods was conducted. Additionally, the new method was applied in machine learning to train some neural network models. The results demonstrate the validity of the proposed method.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMathematics Subject Classification\u003c/strong\u003e: 65K05, 90C30, 90C56\u003c/p\u003e","manuscriptTitle":"A modified version of RMIL conjugate gradient method and its application with backpropagation in machine learning","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-03-20 06:26:02","doi":"10.21203/rs.3.rs-9086018/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-04-20T00:36:39+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-03-25T20:20:37+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"336386868713292455724453265066529941522","date":"2026-03-18T17:26:55+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-03-18T01:02:57+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-03-11T20:45:04+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-03-11T02:19:01+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Inequalities and Applications","date":"2026-03-10T15:48:41+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-inequalities-and-applications","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jiap","sideBox":"Learn more about [Journal of Inequalities and Applications](http://journalofinequalitiesandapplications.springeropen.com)","snPcode":"13660","submissionUrl":"https://submission.nature.com/new-submission/13660/3","title":"Journal of Inequalities and Applications","twitterHandle":"@SpringerMath","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Open","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"e80bdc88-2b38-4cb8-a615-0761a7f4df01","owner":[],"postedDate":"March 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-05-04T04:54:31+00:00","versionOfRecord":[],"versionCreatedAt":"2026-03-20 06:26:02","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-9086018","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-9086018","identity":"rs-9086018","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}