A generalized proximal point algorithm for the nonlinear complementarity problem

Issue		RAIRO Oper. Res. Volume 33, Number 4, October-December 1999


Page(s)		447 - 479
DOI		10.1051/ro:1999117

DOI: 10.1051/ro:1999117

RAIRO Rech. Opér. (vol. 33, n $^\circ$ 4, 1999, pp. 447-479)

A generalized proximal point algorithm for the nonlinear complementarity problem

Regina S. Burachik¹ and Alfredo N. Iusem²

¹Universidade Federal de Rio de Janeiro, COPPE/UFRJ, Programa de Engenharia de Sistemase Computacao, Caixa Postal 68511, CEP 21945-970, Rio de Janeiro - RJ - Brazil.
²Instituto de Matematica Pura e Aplicada, Estrada Dona Castorina, 110, CEP 22460-320, Rio de Janeiro, RJ, Brazil.

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 $\varphi$ -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 $\varphi$ -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.

Keywords: Nonlinear complementarity problem, proximal point methods, monotone operators.

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