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Existence and Uniqueness of Solution for Linear Complementarity Problem in Contact Mechanics · arXivDesk

2508.05777Aug 7, 2025

Existence and Uniqueness of Solution for Linear Complementarity Problem in Contact Mechanics

Jiamin Xu, Nazli Demirer, Vy Pho, He Zhang, Kaixiao Tian, Ketan Bhaidasna, Robert Darbe, Dongmei Chen

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 $\mathbf{M}$

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Although a unique solution is guaranteed in the Linear complementarity problem (LCP) when the matrix M is positive definite, practical applications often involve cases where 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 M and vector q.

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The paper addresses: Existence and Uniqueness of Solution for Linear Complementarity Problem in Contact Mechanics.
Although a unique solution is guaranteed in the Linear complementarity problem (LCP) when the matrix M is positive definite, practical applications often involve cases where M is only positive semi-definite, leading to multiple possible solutions.