tailieunhanh - Báo cáo hóa học: " Research Article A New Method for Solving Monotone Generalized Variational Inequalities"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A New Method for Solving Monotone Generalized Variational Inequalities | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 657192 20 pages doi 2010 657192 Research Article A New Method for Solving Monotone Generalized Variational Inequalities Pham Ngoc Anh and Jong Kyu Kim Department of Mathematics Kyungnam University Masan Kyungnam 631-701 Republic of Korea Correspondence should be addressed to Jong Kyu Kim jongkyuk@ Received 11 May 2010 Revised 27 August 2010 Accepted 4 October 2010 Academic Editor Siegfried Carl Copyright 2010 P. N. Anh and J. K. Kim. 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. We suggest new dual algorithms and iterative methods for solving monotone generalized variational inequalities. Instead of working on the primal space this method performs a dual step on the dual space by using the dual gap function. Under the suitable conditions we prove the convergence of the proposed algorithms and estimate their complexity to reach an e-solution. Some preliminary computational results are reported. 1. Introduction Let C be a convex subset of the real Euclidean space R F be a continuous mapping from C into R and p be a lower semicontinuous convex function from C into R. We say that a point x is a solution of the following generalized variational inequality if it satisfies F x x - x q x - q x 0 Vx e C GVI where denotes the standard dot product in R . Associated with the problem GVI the dual form of this is expressed as following which is to find y e C such that F x x - y q x - y y 0 Vx e C. DGVI In recent years this generalized variational inequalities become an attractive field for many researchers and have many important applications in electricity markets transportations economics and nonlinear analysis see 1-9 . 2 Journal of Inequalities and Applications It is well known that the interior .