Multi-Step Inertial Hybrid and Shrinking Tseng’s Algorithm with Meir–Keeler Contractions for Variational Inclusion Problems

The main aim of this work is to introduce an implicit general iterative method for approximating a solution of a split variational inclusion problem with a hierarchical optimization problem constraint for a countable family of mappings, which are nonexpansive, in the setting of infinite dimensional Hilbert spaces. Convergence theorem of the sequences generated in our proposed implicit algorithm is obtained under some weak assumptions.

The goal of this study was to show how a modified variational inclusion problem can be solved based on Tseng’s method. In this study, we propose a modified Tseng’s method and increase the reliability of the proposed method. This method is to modify the relaxed inertial Tseng’s method by using certain conditions and the parallel technique. We also prove a weak convergence theorem under appropriate assumptions and some symmetry properties and then provide numerical experiments to demonstrate the convergence behavior of the proposed method. Moreover, the proposed method is used for image restoration technology, which takes a corrupt/noisy image and estimates the clean, original image. Finally, we show the signal-to-noise ratio (SNR) to guarantee image quality.

AbstractThe purpose of this paper is to use a modified proximal point algorithm for solving variational inclusion problem in real Hilbert spaces. It is proven that the sequence generated by the proposed iterative algorithm converges strongly to the common solution of the convex minimization and variational inclusion problems.

<abstract><p>This paper is to analyze the approximation solution of a split variational inclusion problem in the framework of Hilbert spaces. For this purpose, inertial hybrid and shrinking projection algorithms are proposed under the effect of a self-adaptive stepsize which does not require information of the norms of the given operators. The strong convergence properties of the proposed algorithms are obtained under mild constraints. Finally, a numerical experiment is given to illustrate the performance of proposed methods and to compare our algorithms with an existing algorithm.</p></abstract>

<p style='text-indent:20px;'>The paper proposes some new iterative algorithms for solving a split variational inclusion problem involving maximally monotone multi-valued operators in a Hilbert space. The algorithms are constructed around the resolvent of operator and the regularization technique to get the strong convergence. Some stepsize rules are incorporated to allow the algorithms to work easily. An application of the proposed algorithms to split feasibility problems is also studied. The computational performance of the new algorithms in comparison with others is shown by some numerical experiments.</p>

We investigate the split variational inclusion problem in Hilbert spaces. We propose efficient algorithms in which, in each iteration, the stepsize is chosen self-adaptive, and proves weak and strong convergence theorems. We provide numerical experiments to validate the theoretical results for solving the split variational inclusion problem as well as the comparison to algorithms defined by Byrne et al. and Chuang, respectively. It is shown that the proposed algorithms outrun other algorithms via numerical experiments. As applications, we apply our method to compressed sensing in signal recovery. The proposed methods have as a main advantage that the computation of the Lipschitz constants for the gradient of functions is dropped in generating the sequences.

The purpose of this paper is to study the convergence analysis of an intermixed algorithm for finding the common element of the set of solutions of split monotone variational inclusion problem (SMIV) and the set of a finite family of variational inequality problems. Under the suitable assumption, a strong convergence theorem has been proved in the framework of a real Hilbert space. In addition, by using our result, we obtain some additional results involving split convex minimization problems (SCMPs) and split feasibility problems (SFPs). Also, we give some numerical examples for supporting our main theorem.