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Mathematical Programming

, Volume 134, Issue 2, pp 349–364 | Cite as

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

  • Ibtihel Ben Gharbia
  • J. Charles GilbertEmail author
Full Length Paper Series A

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 semismooth 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 paper that this result no longer holds when M is a P-matrix of order ≥ 3, 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.

Keywords

Linear complementarity problem Newton’s method Nonconvergence Nonsmooth function M-matrix P-matrix 

Mathematics Subject Classification (2010)

49J52 49M15 90C33 

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Copyright information

© Springer and Mathematical Optimization Society 2011

Authors and Affiliations

  1. 1.Eddimo, CeremadeUniversity Paris-DauphineParis Cedex 16France
  2. 2.INRIA Paris-RocquencourtLe ChesnayFrance

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