It is shown that a nondegenerate square real matrix M is a P-matrix if and only if, whatever is the real vector q, the Newton-min algorithm does not cycle between two points when it is used to solve the linear complementarity problem 0 x ⊥ (M x + q) 0.Key-words: Linear complementarity problem, semismooth Newton method, Newton-min algorithm, NM-matrix, P-matricity characterization, P-matrix. † INRIA Paris-Rocquencourt, project-team Pomdapi, BP 105, F-78153 Le Chesnay (France); e-mails : Ibtihel. Ben Gharbia@inria.fr, Jean-Charles.Gilbert@inria.fr.
Une caractérisation algorithmique de la P-matricitéRésumé : Nous montrons qu'une matrice réelle carrée non dégénérée M est une P-matrice si, et seulement si, quel que soit le vecteur réel q, l'algorithme de Newton-min ne fait pas de cycle de deux points lorsqu'il est utilisé pour résoudre le problème de complémentarité linéaire 0 x ⊥ (M x + q) 0.
The plain Newton-min algorithm to solve the linear complementarity problem (LCP for short) 0x ⊥ (M x + q) 0 can be viewed as a nonsmooth Newton algorithm without globalization technique to solve the system of piecewise linear equations min(x, M x + 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 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.
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