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 Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix --- The full report.
 Ibtihel Ben Gharbia 1, Jean Charles Gilbert ( ) 1
 (2010)
 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 $\Mmat$-matrix of order~$n$, the algorithm is known to converge in at most $n$ iterations. We show in this paper that this result no longer holds when $M$ is a $\Pmat$-matrix of order~$\geq\nobreak3$, since then the algorithm may cycle. $\Pmat$-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 $\Pmat$-matrix of order~$1$ or~$2$.
 1: ESTIME (INRIA Paris-Rocquencourt) INRIA
 Domain : Mathematics/Optimization and Control
 Keywords: Linear complementarity problem – Newton's method – Nonconvergence – Nonsmooth function – M-matrix – P-matrix
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 inria-00442293, version 4 http://hal.inria.fr/inria-00442293 oai:hal.inria.fr:inria-00442293 From: Jean Charles Gilbert <> Submitted on: Tuesday, 28 December 2010 17:39:35 Updated on: Tuesday, 8 March 2011 11:11:55