A generalized proximal point algorithm for the nonlinear complementarity problem
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.
2 Instituto de Matematica Pura e Aplicada, Estrada Dona Castorina, 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil.
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.
Key words: Nonlinear complementarity problem / proximal point methods / monotone operators.
© EDP Sciences, 1999