The modified inertial relaxed CQ algorithm for solving the split feasibility problems

Abstract

In this work, we propose a new version of inertial relaxed CQ algorithms for solving the split feasibility problems in the frameworks of Hilbert spaces. We then prove its strong convergence by using a viscosity approximation method under some weakened assumptions. To be more precisely, the computation on the norm of operators and the metric projections is relaxed. Finally, we provide numerical experiments to illustrate the convergence behavior and to show the effectiveness of the sequences constructed by the inertial technique.

Strong convergence of an inertial iterative algorithm for variational inequality problem, generalized equilibrium problem, and fixed point problem in a Banach space

“…The term θ n (x nx n-1 ) given in (1.9) is called the inertial term. It plays a crucial role in speeding up the convergence properties of iterative method (1.9); for details see [19][20][21][22][23][24][25][26][27]. It is worth to mention that, if we consider θ n = 0, then iterative method (1.9) becomes Krasnosel'skiǐ-Mann type iterative methods; for details, see [28][29][30].…”

Convergence Theorems for Modified Inertial Viscosity Splitting Methods in Banach Spaces

“…The introduction of the inertial viscosity splitting algorithms sheds new light on inclusion problem. Combined with recent research findings ( [4,13,19,20]), Theorem 1 can be further applied to the fixed-point problem, the split feasibility problem and the variational inequality problem. Indeed, it is an important but unsolved problem to choose the optimal inertia parameters α n in the acceleration algorithm.…”

“…it was proved that x n converges strongly to a point z = Q(u) under some mild conditions, where Q is the sunny nonexpansive retraction. Inertial extrapolation is an important technique to speed up the convergence rate [11][12][13][14]. Recently, the fast-iterative algorithms by using inertial extrapolation studied by some authors [15][16][17].…”

A new relaxed CQ algorithm for solving split feasibility problems in Hilbert spaces and its applications

“…Strong convergence theorem. In this section, motivated by the works of Li and Yao [20] and Suantai et al [31] we propose a Mann-type variant of Algorithm 1 with strong convergence property. For this purpose, we define the following convex and differentiable functions…”

“…We wish to thank the anonymous referees for the thorough analysis and review, their comments and suggestions helped tremendously in improving the quality of this paper and made it suitable for publication. We also would like to thank Professor Prasit Cholamjiak for providing reference [31] and for many useful discussions which helped improving our manuscript.…”