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| HAL: inria-00442293, version 4 |
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| Available versions | v1 (2009-12-19) | v2 (2010-04-11) | v3 (2010-09-27) | v4 (2010-12-28) | v5 (2012-12-18) |
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| Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix --- The full report. |
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Ibtihel Ben Gharbia 1Jean Charles Gilbert 1 |
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| (2010) |
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| 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~$2$. |
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| 1: | ESTIME (INRIA Paris-Rocquencourt) |
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INRIA |
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| Domain | : | Mathematics/Optimization and Control |
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| 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 | |
