{"ID":2885402,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.05777","arxiv_id":"2508.05777","title":"Existence and Uniqueness of Solution for Linear Complementarity Problem in Contact Mechanics","abstract":"Although a unique solution is guaranteed in the Linear complementarity problem (LCP) when the matrix $\\mathbf{M}$ is positive definite, practical applications often involve cases where $\\mathbf{M}$ is only positive semi-definite, leading to multiple possible solutions. However, empirical observations suggest that uniqueness can still emerge under certain structural conditions on the matrix $\\mathbf{M}$ and vector $\\mathbf{q}$. Motivated by an unresolved problem in nonlinear modeling for beam contact in directional drilling, this paper systematically investigates conditions under which a unique solution exists for LCPs with certain positive semi-definite matrices $\\mathbf{M}$. We provide a rigorous proof demonstrating the existence and uniqueness of the solution for this specific case and extend our findings to establish a generalized framework applicable to broader classes of LCPs. This framework enhances the understanding of LCP uniqueness conditions and provides theoretical guarantees for solving real-world problems where positive semi-definite matrices $\\mathbf{M}$ arise.","short_abstract":"Although a unique solution is guaranteed in the Linear complementarity problem (LCP) when the matrix $\\mathbf{M}$ is positive definite, practical applications often involve cases where $\\mathbf{M}$ is only positive semi-definite, leading to multiple possible solutions. However, empirical observations suggest that uniqu...","url_abs":"https://arxiv.org/abs/2508.05777","url_pdf":"https://arxiv.org/pdf/2508.05777v1","authors":"[\"Jiamin Xu\",\"Nazli Demirer\",\"Vy Pho\",\"He Zhang\",\"Kaixiao Tian\",\"Ketan Bhaidasna\",\"Robert Darbe\",\"Dongmei Chen\"]","published":"2025-08-07T18:44:07Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}