Strong convergence of an inertial extrapolation method for a split system of minimization problems

Abstract

In this article, we propose an inertial extrapolation-type algorithm for solving split system of minimization problems: finding a common minimizer point of a finite family of proper, lower semicontinuous convex functions and whose image under a linear transformation is also common minimizer point of another finite family of proper, lower semicontinuous convex functions. The strong convergence theorem is given in such a way that the step sizes of our algorithm are selected without the need for any prior information about the operator norm. The results obtained in this article improve and extend many recent ones in the literature. Finally, we give one numerical example to demonstrate the efficiency and implementation of our proposed algorithm.

Mann-Type Inertial Subgradient Extragradient Rules for Variational Inequalities and Common Fixed Points of Nonexpansive and Quasi-Nonexpansive Mappings

“…For N = 1, we now present Lipschitz continuous and pseudomonotone mapping A, quasi-nonexpansive mapping T and nonexpansive mapping T 1 such that Ω = Fix(T 1 ) ∩ Fix(T) ∩ VI(C, A) = ∅. Indeed, let A, T, T 1 : H → H be defined as (Ax)(t) := max{0, x(t)}, (T 1 x)(t) := 1 2 x(t) − 1 2 sin x(t) and (Tx)(t) := 1 2 x(t) + 1 2 sin x(t) for all x ∈ H. It can be easily verified (see, e.g., [8,9]) that A is monotone and L-Lipschitz continuous with L = 1, and the solution set of the VIP for A is given by VI(C, A) = {0} = ∅.…”

“…If VI(C, A) = ∅, one knows that this method has only weak convergence, and only requires that A is monotone and L-Lipschitzian. The literature on the VIP is vast, and Korpelevich's extragradient method has received great attention from many authors, who improved it via various approaches so that some new iterative methods happen to solve the VIP and related optimization problems; see, e.g., [2][3][4][5][6][7][8][9][10][11][12] and the references therein, to name but a few.…”

Image deblurring using a projective inertial parallel subgradient extragradient-line algorithm of variational inequality problems

“…Many problems in the real-world can formulated in the form of the VIP (1), such as economics, engineering mechanics, signal processing, image recovery, transportation, and others (see, for example [3,6,9,13,14]). Several numerical methods have been constructed for solving variational inequalities and related optimization problems, in [1,2,15,20,22,[25][26][27][28][29] and the references cited therein.…”

“…where λ ∈ (0, 1 L ) and P C denotes the metric projection from H onto C. This method converges if B is L-Lipschitz continuous and monotone operator.…”

“…where λ ∈ (0, 1 L ). The subgradient extragradient method for solving VIP (1) has received great attention by many authors (see, e.g., [7,18] and the references therein).…”