Modified splitting algorithms for approximating solutions of split variational inclusions in Hilbert spaces

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.

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

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.

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.

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.

We introduce an iterative method to approximate a common solution of split variational inclusion problem and fixed point problem for nonexpansive semigroups with a way of selecting the stepsizes which does not need any prior information about the operator norms in Hilbert spaces. We prove that the sequences generated by the proposed algorithm converge strongly to a common element of the set of solutions of a split variational inclusion and the set of common fixed points of one-parameter nonexpansive semigroups. Moreover, numerical results demonstrate the performance and convergence of our result, which may be viewed as a refinement and improvement of the previously known results announced by many other researchers.

In this paper, the split variational inclusion problem (SVIP) and the system of equilibrium problems (EP) are considered in Hilbert spaces. Inspired by the works of Byrne et al., López et al., Moudafi and Thukur, Sobumt and Plubtieng, Sitthithakerngkiet et al. and Eslamian and Fakhri, a new self-adaptive step size algorithm is proposed to find a common element of the solution set of the problems SVIP and EP. Convergence theorems are established under suitable conditions for the algorithm and application to the common solution of the fixed point problem, and the split convex optimization problem is considered. Finally, the performances and computational experiments are presented and a comparison with the related algorithms is provided to illustrate the efficiency and applicability of our new algorithms.

The main purpose of this paper is to introduce a viscosity-type iterative algorithm for approximating a common solution of a split variational inclusion problem and a fixed point problem. Using our algorithm, we state and prove a strong convergence theorem for approximating a common solution of a split variational inclusion problem and a fixed point problem for a multivalued quasi-nonexpansive mapping between a Hilbert space and a Banach space. Furthermore, we applied our results to study a split convex minimization problem. Also, a numerical example of our result is given. Our results extend and improve the results of Byrne et al. (J. Nonlinear Convex Anal. 13, 759–775, 2012), Moudafi (J. Optim. Theory Appl. 150, 275–283, 2011), Takahashi and Yao (Fixed Point Theory Appl. 2015, 87, 2015), and a host of other important results in this direction.

In this paper, we first study a hierarchical problem of Baillon’s type, and we study a strong convergence theorem of this problem. For the special case of this convergence theorem, we obtain a strong convergence theorem for the ergodic theorem of Baillon’s type. Our result of the ergodic theorem of Baillon’s type improves and generalizes many existence theorems of this type of problem. Two numerical examples are given to demonstrate our results. As applications of our convergence theorem of the hierarchical problem, we study the unique solution for the following problems: mathematical programming with multiply sets split variational inclusion and fixed point set constraints; mathematical programming with multiple sets split variational inequalities and fixed point set constraints; the variational inequality problem with a system of mixed type equilibria and fixed point set constraints; the variational inequality problem with multiple sets split system of mixed type equilibria and fixed point set constraints; mathematical programming with a system of mixed type equilibria and fixed point set constraints. We give iteration processes for these types of problems and establish the strong convergence for the unique solution of these problems. For our special case, our results can be reduced to the following problems: the unique minimal norm solution of the multiply sets split monotonic variational inclusion problems; the minimum norm solutions for the multiple sets split system of mixed type equilibria problem; the minimum norm solution of the system of mixed type equilibria problem. Our results will have many applications in diverse fields of science.

<abstract><p>In this paper, we present self-adaptive inertial iterative algorithms involving Yosida approximation to investigate a split variational inclusion problem (SVIP) and common solutions of a fixed point problem (FPP) and SVIP in Hilbert spaces. We analyze the weak convergence of the proposed iterative algorithm to explore the approximate solution of the SVIP and strong convergence to estimate the common solution of the SVIP and FPP under some mild suppositions. A numerical example is demonstrated to validate the theoretical findings, and comparison of our iterative methods with some known schemes is outlined.</p></abstract>

In this paper, we present a multi-step hybrid iterative method. It is proven that under appropriate assumptions, the proposed iterative method converges strongly to a common element of fixed point of a finite family of nonexpansive mappings, the solution set of split monotone variational inclusion problem and the solution set of triple hierarchical variational inequality problem (THVI) in real Hilbert spaces. In addition, we give a numerical example of a triple hierarchical system derived from our generalization.

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 purpose of this paper is to introduce a general viscosity implicit iterative method for finding a solution of a split variational inclusion problem (SVIP) with a hierarchical variational inequality (HVI) constraint for a countable family of nonexpansive mappings in Hilbert spaces. Strong convergence theorem is obtained under some mild assumptions.

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.

ABSTRACTIn this work, we consider a nonlinear resolvent integro-differential evolution inclusions in Hilbert spaces. This paper deals with the approximate controllability for nonlinear resolvent integro-differential inclusions in Hilbert spaces. We use Bohnenblust–Karlin's fixed-point theorem to establish a set of sufficient conditions for the approximate controllability for nonlinear resolvent integro-differential inclusions in Hilbert spaces. Further, we extend the result to study the approximate controllability concept with non-local conditions. An example is presented to demonstrate the obtained theory.