Fast hybrid iterative schemes for solving variational inclusion problems

Abstract

Tseng's forward‐backward‐forward splitting method for finding zeros of the sum of Lipschitz continuous monotone and maximal monotone operators is known to converge weakly in infinite dimensional Hilbert spaces. The inertial and viscosity approximation techniques are the techniques widely used to accelerate iterative algorithms and obtain strong convergence, respectively. In this paper, we propose two fast, strongly convergent modifications of Tseng's method and present some consequences and applications of our results. Moreover, we illustrate the performance and computability of our algorithms with relevant numerical examples. The two hybrid techniques incorporate the inertial and viscosity techniques. Unlike the conventional inertial‐viscosity hybrid techniques in the literature, our new algorithms compute both the inertial extrapolation and viscosity approximation simultaneously at the first step of each iteration.

A Regularized Tseng Method for Solving Various Variational Inclusion Problems and Its Application to a Statistical Learning Model

“…and using Lemma 2, we arrive at the conclusion that ū solves (6). The uniqueness of the solution to F x * , u † − x * ≥ 0 according to Lemma 3 implies that ū = x * .…”

“…See, for example, [1][2][3] and references therein. In addition, under some assumptions, such problems involve many important concepts in applied mathematics, such as convex minimization, split feasibility, fixed points, saddle points, and variational inequalities; see, for example, [4][5][6][7].…”

Halpern-Type Inertial Iteration Methods with Self-Adaptive Step Size for Split Common Null Point Problem

“…It is found that this kind of scheme has an improved convergence rate and therefore this scheme was adopted, altered and implemented to solve various nonlinear problems, see, e.g., [23][24][25][26][27][28]. Very recently, Reich and Taiwo [29] studied some fast iterative methods for estimating the solution of variational inclusion problem in which they jointly compute the viscosity approximation and inertial extrapolation in the first step of iterations.…”