A Newton-type Method for Non-smooth Under-determined Systems of Equations

A Newton-type Method for Non-smooth Under-determined Systems of Equations | 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A Newton-type Method for Non-smooth Under-determined Systems of Equations Titus Pinta This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6529210/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 06 Sep, 2025 Read the published version in Numerical Algorithms → Version 1 posted 9 You are reading this latest preprint version Abstract We study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. The notion of regularity employed in our work is based on Newton differentiability, which generalizes semi-smoothness. The classic notion of Newton differentiability does not suffice for our purpose, due to the existence of multiple zeros and as such we extend it to uniform Newton differentiability. In this context, we can show that the distance between the iterates and the set of zeros of the system decreases super-linearly. For the special case of smooth equations, the assumptions of our algorithm are simplified. Finally, we provide some numerical examples to showcase the behavior of our proposed method. The key example is a toy model of complementarity constraint problems, showing that our method has great application potential across engineering fields. MSC Classification: 90C53 , 49M15 Newton’s Method Under-determined systems Higher Order Methods Nonsmooth Equations Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 06 Sep, 2025 Read the published version in Numerical Algorithms → Version 1 posted Editorial decision: Revision requested 18 Jun, 2025 Reviews received at journal 16 Jun, 2025 Reviewers agreed at journal 21 May, 2025 Reviews received at journal 19 May, 2025 Reviewers agreed at journal 04 May, 2025 Reviewers invited by journal 30 Apr, 2025 Editor assigned by journal 30 Apr, 2025 Submission checks completed at journal 30 Apr, 2025 First submitted to journal 25 Apr, 2025 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. 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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-6529210","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":451647183,"identity":"a0fd9c81-238d-48d1-885f-41e5ae328c02","order_by":0,"name":"Titus Pinta","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABD0lEQVRIie3Pv0sDMRTA8VcCueXd3ZpQaf+FB4WT4nD/iiA4OQhdbgoBIVtx9s/o5pgSqEv/gcNC63KTgyJICwpGa38MF2eHfOEucNyH9wIQi/3H2PZA/3RWh88EkOh2gYz9Eg7M/wdiT9AGyG6MJ1wcCIRJmeQzuYbFyWn/pqle75XKdTp9q64VYLfd+MV4F6HBoeFFfTd3QtjsQs7JAWbnYQLgkGZQPKbGClg+k9RkocTg9blf7Jsk76NPo0Tf4mCjyS/2BxH4Q7BgHcMEWSz8FBYmjg3OkBpPrkZybJyc2OxyqMlhiCS306d6XS1Kcg+Tl41Rec+mrtYfqhci2/xljycfvcO1kFgsFovt+gJBk0xKirk+VAAAAABJRU5ErkJggg==","orcid":"","institution":"École Nationale Supérieure de Techniques Avancées","correspondingAuthor":true,"prefix":"","firstName":"Titus","middleName":"","lastName":"Pinta","suffix":""}],"badges":[],"createdAt":"2025-04-25 13:08:31","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6529210/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6529210/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s11075-025-02212-8","type":"published","date":"2025-09-06T15:57:40+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":90828120,"identity":"4c359906-df3f-4885-9d4c-9e803deb0963","added_by":"auto","created_at":"2025-09-08 16:05:53","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":351613,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6529210/v1_covered_74ed0f89-6214-452d-9157-445cc68591b5.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"A Newton-type Method for Non-smooth Under-determined Systems of Equations","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"numerical-algorithms","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"numa","sideBox":"Learn more about [Numerical Algorithms](http://link.springer.com/journal/11075)","snPcode":"11075","submissionUrl":"https://submission.nature.com/new-submission/11075/3","title":"Numerical Algorithms","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Newton’s Method, Under-determined systems, Higher Order Methods, Nonsmooth Equations","lastPublishedDoi":"10.21203/rs.3.rs-6529210/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6529210/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe study a variant of Newton's algorithm applied to under-determined systems of non-smooth equations. 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