Convergence of Hybrid Steepest–Descent Methods for Generalized Variational Inequalities

Abstract

Abstract. In this paper we consider the general variational inequality GVI(F, g, C) where F and g are mappings from a Hilbert space into itself and C is the fixed points set of a nonexpansive mapping. We propose two iterative algorithms to find approximate solutions of the GVI (F, g, C). Strong convergence results are established and applications to constrained generalized pseudoinverse are included.

Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces

“…Further, the hybrid steepest-descent method has been widely extended to develop various algorithms for finding solutions of the VI(F , C ); see, e.g., [11][12][13]. In particular, very recently, Zeng, Wong and Yao [12] introduced and studied the following modified hybrid steepest-descent Algorithms 1.1 and 1.2 with variable parameters for computing approximate solutions of the VI(F , C ).…”

“…In particular, very recently, Zeng, Wong and Yao [12] introduced and studied the following modified hybrid steepest-descent Algorithms 1.1 and 1.2 with variable parameters for computing approximate solutions of the VI(F , C ). Algorithm 1.1 ([12]).…”

A viscosity of Cesaro mean approximation method for split generalized equilibrium, variational inequality and fixed point problems

“…It is well known that if A is strongly monotone and Lipschitz continuous mapping on C then VIP (1.2) has a unique solution. There are several different approaches towards solving this problem in finite dimensional and infinite dimensional spaces see [6,7,8,14,16,20,31,35,40] and the research in this direction is intensively continued.…”

Hybrid Steepest Descent Method with Variable Parameters for General Variational Inequalities

“…Several numerical methods including the projection and its variant forms, Wiener-Hofp equations, auxiliary principle, and descent type have been developed for solving the variational inequalities and related optimization problems. The reader is referred to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the references therein.…”

“…It is clear that it can not be directly applied to computing solution of the GVI(F,g,C) due to the presence of g. Therefore, an important problem is how to apply hybrid steepest descent method to solving GVI(F,g,C). For this purpose, Zeng et al [13] introduced a hybrid steepest descent method for solving the GVI(F,g,C) as follows. …”