Existence and Uniqueness of Solution for Linear Complementarity Problem in Contact Mechanics
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arXiv
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| Autori principali: | , , , , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915434324819968 |
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| author | Xu, Jiamin Demirer, Nazli Pho, Vy Zhang, He Tian, Kaixiao Bhaidasna, Ketan Darbe, Robert Chen, Dongmei |
| author_facet | Xu, Jiamin Demirer, Nazli Pho, Vy Zhang, He Tian, Kaixiao Bhaidasna, Ketan Darbe, Robert Chen, Dongmei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_05777 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and Uniqueness of Solution for Linear Complementarity Problem in Contact Mechanics Xu, Jiamin Demirer, Nazli Pho, Vy Zhang, He Tian, Kaixiao Bhaidasna, Ketan Darbe, Robert Chen, Dongmei Optimization and Control 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. |
| title | Existence and Uniqueness of Solution for Linear Complementarity Problem in Contact Mechanics |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2508.05777 |