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A generalized proximal point algorithmfor the nonlinear complementarity problem

Published online by Cambridge University Press:  15 August 2002

Regina S. Burachik
Affiliation:
Universidade Federal de Rio de Janeiro, COPPE/UFRJ, Programa de Engenharia de Sistemase Computacao, Caixa Postal 68511, CEP 21945-970, Rio de Janeiro - RJ - Brazil.
Alfredo N. Iusem
Affiliation:
Instituto de Matematica Pura e Aplicada, Estrada Dona Castorina, 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil.
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Abstract

We consider a generalized proximal point method (GPPA) for solving the nonlinear complementarity problem with monotone operators in Rn. It differs from the classical proximal point method discussed by Rockafellar for the problem of finding zeroes of monotone operators in the use of generalized distances, called φ-divergences, instead of the Euclidean one. These distances play not only a regularization role but also a penalization one, forcing the sequence generated by the method to remain in the interior of the feasible set, so that the method behaves like an interior point one. Under appropriate assumptions on the φ-divergence and the monotone operator we prove that the sequence converges if and only if the problem has solutions, in which case the limit is a solution. If the problem does not have solutions, then the sequence is unbounded. We extend previous results for the proximal point method concerning convex optimization problems.

Type
Research Article
Copyright
© EDP Sciences, 1999

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