Fixed Point Theory and Applications
Volume 2007 (2007), Article ID 76040, 15 pages
doi:10.1155/2007/76040
Research Article
An Algorithm Based on Resolvant Operators for Solving Positively Semidefinite Variational Inequalities
Department of Applied Mathematics, Dalian University of Technology, Dalian, Liaoning 116024, China
Received 16 June 2007; Accepted 19 September 2007
Academic Editor: Nan-Jing Huang
Copyright © 2007 Juhe Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
A new monotonicity, M-monotonicity, is introduced, and the resolvant operator of an M-monotone operator is proved to be single-valued and Lipschitz continuous. With the help of the resolvant operator, the positively semidefinite general variational inequality
(VI) problem VI (S+n,F+G) is transformed into a fixed point problem of a nonexpansive mapping. And a proximal point algorithm is constructed to solve the fixed point problem, which is proved to have a global convergence under the condition that F in the VI problem is strongly monotone and Lipschitz continuous. Furthermore, a convergent path Newton method is given for calculating ε-solutions to the sequence of fixed point problems, enabling the
proximal point algorithm to be implementable.