Levitin–Polyak well-posedness of split multivalued variational inequalities

In this paper, we propose a split nonconvex variational inequality problem which is a natural extension of split convex variational inequality problem in two different Hilbert spaces. Relying on the prox-regularity notion, we introduce and establish the convergence of an iterative method for the new split nonconvex variational inequality problem. Further, we also establish the convergence of an iterative method for the split convex variational inequality problem. The results presented in this paper are new and different form the previously known results for nonconvex (convex) variational inequality problems. These results also generalize, unify, and improve the previously known results of this area.

The purpose of this paper is to investigate Levitin–polyak well-posedness by perturbations of the split variational inequality problem in reflexive Banach spaces. Furi-Vignoli-type characterizations are established for the well-posedness. We prove that the weak generalized Levitin–Polyak well-posedness by perturbations is equivalent to the nonemptiness and boundedness of the solution set of the problem. Finally, we discuss the relations between the Levitin–Polyak well-posedness by perturbations of the split variational inequality problem and the Levitin–Polyak well-posedness by perturbations of the split minimization problem when the split variational inequality problem arises from the split minimization problem.

In this paper, we introduce a split general mixed variational inequality problem, which is a natural extension of a split variational inequality problem, split general quasi-variational inequality problem in Hilbert spaces. Using the resolvent operator technique, we propose a perturbed iterative algorithm for the split general mixed variational inequality problem. Further, we discuss the convergence criteria of the iterative algorithm. The results presented in this paper extend and improve many previously known results in this area.

We introduce a split general strong nonlinear quasi-variational inequality problem which is a natural extension of a split general quasi-variational inequality problem, split variational inequality problem, and quasi-variational and variational inequality problems in Hilbert spaces. Using the projection method, we propose an iterative algorithm for the split general strongly nonlinear quasi-variational inequality problem and discuss the convergence criteria of the iterative algorithm. The results presented here generalized, unify, and improve many previously known results for quasi-variational and variational inequality problems.

The purpose of this paper is to study variational inequality problem over the solution set of multiple-set split monotone variational inclusion problem. We propose an iterative algorithm with inertial method for finding an approximate solution of this problem in real Hilbert spaces. Strong convergence of the sequence of iterates generated from the proposed method is obtained under some mild assumptions. The iterative scheme does not require prior knowledge of operator norm. Also we present some applications of our main result to solve the bilevel programming problem, the bilevel monotone variational inequalities, the split minimization problem, the multiple-set split feasibility problem and the multiple set split variational inequality problem.

In this paper, we first consider a split variational inclusion problem and give several strong convergence theorems in Hilbert spaces, like the Halpern-Mann type iteration method and the regularized iteration method. As applications, we consider the algorithms for a split feasibility problem and a split optimization problem and give strong convergence theorems for these problems in Hilbert spaces. Our results for the split feasibility problem improve the related results in the literature. MSC:47H10, 49J40, 54H25.

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 purpose of this paper is to introduce and study a general split variational inclusion problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split variational inclusion problem. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces.

This paper proposes a new self-adaptive algorithm for solving the multiple-set split variational inequality problem in Hilbert spaces. Our algorithm uses dynamic step-sizes, chosen based on information of the previous step. In comparison with the work by Censor et al. [Numer Algorithms. 2012;59:301–323], the new algorithm gives strong convergence results and does not require information about the transformation operator's norm. Some applications of our main results regarding the solution of the multiple-set split feasibility problem and the split feasibility problem are presented and show that the iterative method converges strongly under weaker assumptions than the ones used recently by Xu [Inverse Probl. 2006;22:2021–2034] and by Buong [Numer Algorithms. 2017;76:783–798]. Numerical experiments on finite-dimensional and infinite-dimensional spaces and an application to discrete optimal control problems are reported to demonstrate the advantages and efficiency of the proposed algorithms over some existing results.

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.

<p style='text-indent:20px;'>In this paper, we present extension of a class of split variational inequality problem and fixed point problem due to Lohawech et al. (J. Ineq Appl. 358, 2018) to a class of multiple sets split variational inequality problem and common fixed point problem (CMSSVICFP) in Hilbert spaces. Using the Halpern subgradient extragradient theorem of variational inequality problems, we propose a parallel Halpern subgradient extragradient CQ-method with adaptive step-size for solving the CMSSVICFP. We show that a sequence generated by the proposed algorithm converges strongly to the solution of the CMSSVICFP. We give a numerical example and perform some preliminary numerical tests to illustrate the numerical efficiency of our method.</p>

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.

Strong convergence theorem of viscosity approximation methods for nonexpansive mapping have been studied. We also know that CQ algorithm for solving the split feasibility problem (SFP) has a weak convergence result. In this paper, we use viscosity approximation methods and some related knowledge to solve a class of generalized SFP’s with monotone variational inequalities in Hilbert space. We propose some iterative algorithms based on viscosity approximation methods and get strong convergence theorems. As applications, we can use algorithms we proposed for solving split variational inequality problems (SVIP), split constrained convex minimization problems and some related problems in Hilbert space.

In this paper, we propose and investigate two new iterative algorithms for solving the split equality variational inclusion problem in Hilbert spaces. We also prove that the sequences generated by the proposed algorithms converge strongly to a common solution of the split equality variational inclusion problem and fixed points of a family of nonexpansive mappings, which is also an unique solution of a variational inequality as an optimality condition for a minimization problem. The results presented in this paper extend and generalize a variety of existing results in this area.

In this paper, we introduce a new iterative algorithm for approximating a common solution of certain class of multiple-sets split variational inequality problems. The sequence of the proposed iterative algorithm is proved to converge strongly in Hilbert spaces. As application, we obtain some strong convergence results for some classes of multiple-sets split convex minimization problems.