A splitting algorithm with inertial and correction terms for solving split variational inclusion problems and its applications

Split variational inclusions encompass a broad category of problems, incorporating several previously known split-type issues such as split feasibility, split zero problems, split variational inequalities and so on. This problem can be applied to solve real-world problems in engineering, sciences, medicine and so on. In this paper, we present splitting algorithms with linearization for solving the split variational inclusion problem in Hilbert spaces. We develop the algorithm proposed by Dong et al. (An alternated inertial general splitting method with linearization for the split feasibility problem. Optimization 72(10):2585–2607) by using the inertial technique and extending the result from split feasibility problem to generalized split variational inclusion problem. Based on the self-adaptive stepsize, we introduce and analyse a new splitting algorithm for solving the problem without the Lipschitz condition. Under suitable assumptions, we prove that the sequence generated by our main iterative algorithm converges weakly to a solution. Finally, we illustrate numerical performance of the proposed algorithm and give applications to the split feasibility problem which can be applied to a compressed sensing in signal recovery.

ABSTRACTIn this article, we study the generalized split variational inclusion problem. For this purpose, motivated by the projected Landweber algorithm for the split equality problem, we first present a simultaneous subgradient extragradient algorithm and give related convergence theorems for the proposed algorithm. Next, motivated by the alternating CQ-algorithm for the split equality problem, we propose another simultaneous subgradient extragradient algorithm to study the general split variational inclusion problem. As applications, we consider the split equality problem, split feasibility problem, split variational inclusion problem, and variational inclusion problem in Hilbert spaces.

The split feasibility problem is an inverse problem which arises in signal processing and medical image reconstruction. So there is practical value in studying it. While both the split equality problem and the split variational inclusion problem are generalized form of the split feasibility problem which are more meaningful than the split feasibility problem. In this paper, fusing the two problems, we research a split inclusion problem and propose relevant methods for solving it. What counts is that not only the proposed algorithms have strong convergence, but also the limit points of the algorithms are the minimal norm solution of the split inclusion problem.

In this study, we address split variational inclusion problems in Hilbert spaces and introduce a new algorithm that combines multi-inertial extrapolation with a self-adaptive stepsize technique. The method is designed to enhance convergence performance while relaxing typical parameter constraints. Under suitable assumptions, we establish a weak convergence theorem. To validate the effectiveness of the approach, we apply it to real-world problems in image restoration and medical data classification. \begin{keyword}Split variational inclusion problems \sep multi-inertial extrapolation \sep weak convergence \sep image restoration \sep data classification. \MSC{47H09 \sep 47H10 \sep 47J05 \sep 47J25 \sep 49J40 \sep 35A15

ABSTRACTIn convex optimization, numerous problems in applied sciences can be modelled as the split variational inclusion problem (SVIP). In this connection, we aim to design new and efficient proximal type algorithms which are based on the inertial technique and the linesearches terminology. We then discuss its convergence under some suitable conditions without the assumption on the operator norm. We also apply our main result to the split minimization problem, the split feasibility problem, the relaxed split feasibility problem and the linear inverse problem. Finally, we provide some numerical experiments and comparisons to these problems. The obtained result mainly improves the recent results investigated by Chuang.

In this paper, we present a proximal split feasibility algorithm with an additional inertial extrapolation term for solving a proximal split feasibility problem under weaker conditions on the step sizes. The two convex and lower semi continuous objective functions are assumed to be non-smooth. Some applications to split inclusion problem and split equilibrium problem are given. We demonstrate the efficiency of the proposed algorithm with numerical experiments.

This paper designs and studies a numerically fast proximal point algorithm to solve the monotone inclusion problem in Hilbert spaces. Our first proposed algorithm combines the proximal point algorithm, inertial term, and two correction terms. The addition of two correction terms is considered to further numerically accelerate the convergence speed of the inertial proximal point algorithm already studied in the literature. In our convergence results, we obtain both weak and linear convergence of the first proposed algorithm under some standard assumptions. Furthermore, we modify the first algorithm to obtain a strongly convergent algorithm. Applications of our proposed algorithm to mixed variational inequalities, strongly quasi-convex minimization problems, and Douglas–Rachford splitting algorithm are given. Numerical results show that our proposed algorithm outperforms other related algorithms in the literature.

In this paper, we study and introduce a self adaptive method to­gether with a Halpern iterative algorithm for approximating solutions of multiple-sets split monotone variational inclusion problem which in­cludes the multiple-sets split feasibility problem, split feasibility prob­lem, split monotone variational inclusion problem and split variational inclusion problem, to mention a few. Using our iterative algorithm, we prove a strong convergence result for approximating the solution of the aforementioned problems. Numerical examples on finite-dimensional and infinite-dimensional spaces are displayed to illustrate the perfor­mance of our iterative method. The result discussed in this article extends and complements many related results in literature.

Alicia Cordero and Juan R. Torregrosa were partially supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02.

The aim of this paper is to introduce a modified viscosity iterative method to approximate a solution of the split variational inclusion problem and fixed point problem for a uniformly continuous multivalued total asymptotically strictly pseudocontractive mapping in C A T ( 0 ) spaces. A strong convergence theorem for the above problem is established and several important known results are deduced as corollaries to it. Furthermore, we solve a split Hammerstein integral inclusion problem and fixed point problem as an application to validate our result. It seems that our main result in the split variational inclusion problem is new in the setting of C A T ( 0 ) spaces.

In this paper, we introduced an inertial extragradient algorithm to approximate the common solution of split fixed point, split variational inclusion and split equilibrium problems involving nonexpansive mappings and pseudomonotone Lipschitz-type bifunctions in Hilbert spaces. Moreover, using some assumptions on the control parameters, we prove the strong convergence of the proposed algorithm and then apply our main result to solve the split minimization, split feasibility and split variational inequality problems. We also present some numerical examples to show the effectiveness and applicability of the proposed scheme. We include tables illustrating the number of iterations, the CPU time for convergence, comparisons among different algorithms, and the error analysis. We apply our proposed scheme to solve the image restoration problem as another application of the result presented herein.

If S and T are two non-self-mappings, then a solution of equation Sa*=Ta*=a* does not necessarily exist. The common best proximity point problem is to find the approximate optimal solution of such type of equation and have a key role in theory of approximation and optimization. The primary goal of this paper is to introduce an inertial-type self-adaptive algorithm for solving the common best proximity point, generalized equilibrium and split variational inclusion problems in Hilbert spaces. The strong convergence of the proposed algorithm is given under some mild conditions. It is worth mentioning that the step size in many existing algorithms requires the prior knowledge of operator norms which is difficult to compute, whereas our proposed algorithm does not require this condition. Numerical examples are given to illustrate the efficiency and applicability of the proposed approach. We further apply the proposed algorithm to an image restoration problem and show that it achieves a higher signal-to-noise ratio compared with the existing algorithms considered in this study.

A hybrid viscosity algorithm via modify the hybrid steepest descent method for solving the split variational inclusion in image reconstruction and fixed point problems

The split variational inclusion problem is an important problem, and it is a generalization of the split feasibility problem. In this paper, we present a descent-conjugate gradient algorithm for the split variational inclusion problems in Hilbert spaces. Next, a strong convergence theorem of the proposed algorithm is proved under suitable conditions. As an application, we give a new strong convergence theorem for the split feasibility problem in Hilbert spaces. Finally, we give numerical results for split variational inclusion problems to demonstrate the efficiency of the proposed algorithm.

In this paper, let the BPVIP, GEPS, and SCFPP represent a bilevel pseudomonotone variational inequality problem, a generalized equilibrium problems system, and a split common fixed point problem involving demimetric mappings in real Hilbert spaces, respectively. We devise a composite subgradient extragradient rule with an inertial correction term for solving the BPVIP with constraints of GEPS and SCFPP, where the rule exploits the inertial technique with a correction term and a self-adaptive stepsize strategy. The BPVIP consists of the upper-level VIP for one strongly monotone operator and the lower-level VIP for another pseudomonotone operator. The strong convergence result for the designed algorithm is established under certain suitable conditions. In addition, the main result is employed to handle a bilevel split pseudomonotone variational inequality problem (BSPVIP). Lastly, an illustrated instance is utilized to back up the applicability and performability of the suggested rule.