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Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix.

Abstract : The plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) $0\leq x\perp(Mx+q)\geq0$ can be viewed as a nonsmooth Newton algorithm without globalization technique to solve the system of piecewise linear equations $\min(x,Mx+q)=0$, which is equivalent to the LCP. When $M$ is an $M$-matrix of order~$n$, the algorithm is known to converge in at most $n$ iterations. We show in this note that this result no longer holds when $M$ is a $P$-matrix of order~$\geq\nobreak3$, since then the algorithm may cycle. $P$-matrices are interesting since they are those ensuring the existence and uniqueness of the solution to the LCP for an arbitrary~$q$. Incidentally, convergence occurs for a $P$-matrix of order~$1$ or~$2$.
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https://hal.inria.fr/inria-00442293
Contributor : Jean Charles Gilbert <>
Submitted on : Sunday, April 11, 2010 - 12:27:24 PM
Last modification on : Friday, May 25, 2018 - 12:02:03 PM
Long-term archiving on: : Thursday, September 23, 2010 - 12:48:10 PM

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  • HAL Id : inria-00442293, version 2

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Ibtihel Ben Gharbia, Jean Charles Gilbert. Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix.. [Research Report] RR-7160, 2010, pp.18. ⟨inria-00442293v2⟩

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