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.