Parameter Regulation and Projection Enhancement Matrix Splitting Method: Efficient Solution for $H_+$-Matrix Linear Complementarity Problems

Parameter Regulation and Projection Enhancement Matrix Splitting Method: Efficient Solution for $H_+$-Matrix Linear Complementarity Problems | 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 Parameter Regulation and Projection Enhancement Matrix Splitting Method: Efficient Solution for $H_+$-Matrix Linear Complementarity Problems Yajun Xie, Yuting Zhang, Jianfeng Li This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9020076/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 11 You are reading this latest preprint version Abstract The Linear Complementarity Problem (LCP) serves as a crucial mathematical model for characterizing scenarios such as Nash equilibria in game theory, supply and demand balance in economic systems, and dynamic traffic flow distribution. The development of efficient numerical solutions for LCP holds significant theoretical value and practical significance. This paper proposes the Modified Matrix Splitting Iteration (MMSI) method, which introduces a double-diagonal parameter matrix and a relaxation factor to construct a novel iteration format. It innovatively integrates a projection operator to enhance the robustness of the initial point. Theoretical proof shows that when the coefficient matrix $M$ is a positive definite $H_+$ matrix with positive diagonal elements, the algorithm globally converges to a unique solution. Numerical experiments fully demonstrate the efficiency and stability of this algorithm. Linear complementarity problem Modified matrix splitting method Projection operator Convergence Numerical test Full Text Additional Declarations No competing interests reported. Supplementary Files Highlights.docx DeclarationofInterestStatement.docx AuthorAgreementStatement.docx Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 06 Apr, 2026 Reviews received at journal 30 Mar, 2026 Reviews received at journal 29 Mar, 2026 Reviews received at journal 28 Mar, 2026 Reviewers agreed at journal 23 Mar, 2026 Reviewers agreed at journal 23 Mar, 2026 Reviewers agreed at journal 23 Mar, 2026 Reviewers invited by journal 22 Mar, 2026 Editor assigned by journal 04 Mar, 2026 Submission checks completed at journal 04 Mar, 2026 First submitted to journal 03 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. 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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-9020076","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":610582403,"identity":"bce65a29-ca84-4c19-acb0-2afb5abe381b","order_by":0,"name":"Yajun 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