ITERATIVE METHOD FOR SOLVING A MINIMUM NORM PROBLEM | Thắng | TNU Journal of Science and Technology

ITERATIVE METHOD FOR SOLVING A MINIMUM NORM PROBLEM

About this article

Received: 23/09/19                Revised: 23/10/19                Published: 27/11/19

Authors

Nguyen Tat Thang Email to author, Thai Nguyen University

Abstract


In this paper we study the problem of finding a minimum norm solution over the set of
solutions of a variational inequality in Hilbert spaces. In order to solve this bilevel problem, we
propose a new iterative method and establish a strong convergence theorem for it.


Keywords


Variational inequality; Hilbert space; minimum norm; bilevel problem; monotone operator.

References


[1]. Y. Censor, A. Gibali, S. Reich, "The subgradient extragradient method for solving variational inequalities in Hilbert space", J. Optim. Theory Appl. , 148(2), pp. 318–335, 2011.

[2]. R. Glowinski, Numerical Methods for Nonlinear Variational Problems, Springer, New York, NY, 1984.

[3]. K. Goebel, W.A. Kirk, Topics on metric fixed point theory, Cambridge University Press, Cambridge, England, 1990.

[4]. P. Jaillet, D. Lamberton, B. Lapeyre, “Variational Inequalities and the Pricing of American Options”, Acta Applicandae Mathematica, 21, pp. 263–289, 1990.

[5]. D. Kinderlehrer, and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980.

[6]. R. Kraikaew, S. Saejung, "Strong convergence of the Halpern subgradient extra-gradient method for solving variational inequalities in Hilbert spaces", J. Optim. Theory Appl., 163(2), pp. 399–412, 2014.

[7]. Y.V. Malitsky, "Projected reflected gradient methods for monotone variational inequalities.", SIAM J. Optim. , 25(1), pp. 502–520, 2015.

[8]. H.K. Xu, "Iterative algorithms for nonliner operators", J. London Math. Soc. , 66, pp. 240–256, 2002.

[9]. I. Yamada (2001), "The hybrid steepest descent method for the variational inequality problem over the intersection of fixed point sets of nonexpansive mappings", In inherently parallel algorithm for feasibility and optimization and their applications edited by: D. Butnariu, Y. Censor, and S. Reich, Elsevier., 473–504, 2001.

[10]. E. Zeidler, Nonlinear Functional Analysis and Its Applications, III: Variational Methods and Applications, Springer, New York, 1985.


Refbacks

  • There are currently no refbacks.
TNU Journal of Science and Technology
Rooms 408, 409 - Administration Building - Thai Nguyen University
Tan Thinh Ward - Thai Nguyen City
Phone: (+84) 208 3840 288 - E-mail: jst@tnu.edu.vn
Based on Open Journal Systems
©2018 All Rights Reserved