Solution Methods for Pseudomonotone Variational Inequalities |
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Authors: | N N Tam J C Yao N D Yen |
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Institution: | (1) Department of Mathematics, Hanoi University of Pedagogy No. 2, Xuan Hoa, Phuc Yen, Vinh Phuc, Vietnam;(2) Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan;(3) Institute of Mathematics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam |
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Abstract: | We extend some results due to Thanh-Hao (Acta Math. Vietnam. 31: 283–289, 2006]) and Noor (J. Optim. Theory Appl. 115:447–452, 2002]). The first paper established a convergence theorem for the Tikhonov regularization method (TRM) applied to finite-dimensional
pseudomonotone variational inequalities (VIs), answering in the affirmative an open question stated by Facchinei and Pang
(Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer, New York, 2003]). The second paper discussed the application of the proximal point algorithm (PPA) to pseudomonotone VIs. In this paper,
new facts on the convergence of TRM and PPA (both the exact and inexact versions of PPA) for pseudomonotone VIs in Hilbert
spaces are obtained and a partial answer to a question stated in (Acta Math. Vietnam. 31:283–289, 2006]) is given. As a byproduct, we show that the convergence theorem for inexact PPA applied to infinite-dimensional monotone
variational inequalities can be proved without using the theory of maximal monotone operators.
This research was supported in part by a grant from the National Sun Yat-Sen University, Kaohsiung, Taiwan. It has been carried
out under the agreement between the National Sun Yat-Sen University, Kaohsiung, Taiwan and the University of Pisa, Pisa, Italy.
The authors thank the anonymous referee for useful comments and suggestions. |
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Keywords: | Variational inequalities Pseudomonotone operators Tikhonov regularization method Proximal point algorithms Convergence |
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