Hierarchical variational inequality problem and split common fixed point of averaged operators

In this paper, we construct a novel algorithm for the split common fixed point problem for two demicontractive operators in Hilbert spaces. By using inertial self-adaptive algorithms, we obtain strong convergence results for finding a solution of the split common fixed point problems. Applications to solving the split minimization problem and the split feasibility problem are included. Our results extend and generalize many previously known results in this research area. Moreover, numerical experiments are supplied to demonstrate the convergence behavior and efficiency of the proposed algorithm.

In this paper, we first introduce a new algorithm with a viscosity iteration method for solving the split common fixed point problem (SCFP) for a finite family of nonexpansive semigroups. We also present a new algorithm for solving the SCFP for an infinite family of quasi-nonexpansive mappings. We establish strong convergence of these algorithms in an infinite-dimensional Hilbert spaces. As application, we obtain strong convergence theorems for split variational inequality problems and split common null point problems. Our results improve and extend the related results in the literature.

In this paper, we introduce an efficient subgradient extragradient (SE) based method for solving variational inequality problems with monotone operator in Hilbert space. In many existing SE methods, two values of operator are needed over each iteration and the Lipschitz constant of the operator or linesearch is required for estimating step sizes, which are usually not practical and expensive. To overcome these drawbacks, we present an inertial SE based algorithm with adaptive step sizes, estimated by using an approximation of the local Lipschitz constant without running a linesearch. Each iteration of the method only requires a projection on the feasible set and a value of the operator. The numerical experiments illustrate the efficiency of the proposed algorithm.

In this paper, we introduce a new strongly convergent algorithm for solving common fixed point problems in Hilbert spaces for a finite (infinite) family of demicontractive operators. As a consequence, we obtain strong convergence theorems for split common fixed point problems for a finite family of demicontractive operators, split variational inequality problems for inverse strongly monotone operators and split common null point problems for maximal monotone operators. Finally, the performance of the proposed algorithm is also illustrated by several preliminary numerical experiments.

We extend the notion of asymptotically nonexpansive mapping to the more general class, namely, e-enriched asymptotically nonexpansive mappings. It is shown, with an example, that the class of e-enriched asymptoticallynonexpansive mappings is more general than the class of asymptotically nonexpansive mappings. Certain weak and strong convergence theorems are then proved for the iterative approximation of split common fixed point problem involving the class of(e,ϑ)-enriched strictly quasi-pseudocontractive mappings and the class of e-enriched asymptotically nonexpansive mappings in the domain of two Banach spaces. Furthermore, a significant result for the hierarchical variational inequality problem is obtained as a consequence of our main result.

In this paper we consider a new general iterative method for solving the split common fixed point problem and variational inclusion problem. It entails finding a point which belongs to the set of common fixed points of a finite family of demimetric mappings and the common zero point set of a family of monotone operators in a Hilbert space such that its image under a linear transformation belongs to the set of fixed points of a demimetric mapping in a uniformly convex and smooth Banach space in the image space. Strong convergence theorem is established under some suitable conditions. Results presented in this paper may be viewed as a refinement and important generalizations of the previously known results announced by many other authors.

Our contribution in this paper is to propose an iterative algorithm which does not require prior knowledge of operator norm and prove strong convergence theorem for approximating a solution of split common fixed point problem of demicontractive mappings in a real Hilbert space. So many authors have used algorithms involving the operator norm for solving split common fixed point problem, but as widely known the computation of these algorithms may be difficult and for this reason, authors have recently started constructing iterative algorithms with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm. We introduce a new algorithm for solving the split common fixed point problem for demicontractive mappings with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm and then prove strong convergence of the sequence in real Hilbert spaces. Finally, we give some applications of our result and numerical example at the end of the paper.

<abstract><p>The theory of variational inequalities is an important tool in physics, engineering, finance, and optimization theory. The projection algorithm and its variants are useful tools for determining the approximate solution to the variational inequality problem. This paper introduces three distinct extragradient algorithms for dealing with variational inequality problems involving quasi-monotone and semistrictly quasi-monotone operators in infinite-dimensional real Hilbert spaces. This problem is a general mathematical model that incorporates a set of applied mathematical models as an example, such as equilibrium models, optimization problems, fixed point problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithms employ both fixed and variable stepsize rules that are iteratively transformed based on previous iterations. These algorithms are based on the fact that no prior knowledge of the Lipschitz constant or any line-search framework is required. To demonstrate the convergence of the proposed algorithms, some simple conditions are used. Numerous experiments have been conducted to highlight the numerical capabilities of algorithms.</p></abstract>

The present paper deals with the common solution method for finding a fixed point of a nonexpansive mapping and a solution of split hierarchical Minty variational inequality problems. We discuss the weak convergence of the sequences generated by the proposed method to a common solution of a fixed point problem and a split hierarchical Minty variational inequality problem. An example is presented to illustrate the proposed algorithm and result.

In this paper, we investigate a class of hierarchical variational inequalities (HVIPs, i.e., strongly monotone variational inequality problems defined on the solution set of multiple-set split common fixed point problems) with quasi-pseudocontractive mappings in real Hilbert spaces, which special cases can be found in many important engineering practical applications such as image recognizing, signal processing and machine learning. In order to solve HVIPs of potential application value, inspired by the primal-dual algorithm, we propose a novel accelerated cyclic iterative algorithm that combines the inertial method with a correction term and a self-adaptive step-size technique. Our approach eliminates the need for prior knowledge of the bounded linear operator norm. Under appropriate assumptions, we establish strong convergence of the algorithm. Finally, we apply our novel iterative approximation to solve multiple-set split feasibility problems and verify the effectiveness of the proposed iterative algorithm through numerical results.

We introduce an iterative sequence for finding the common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for tree inverse-strongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to finding solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of paper we utilize our results to study the zeros of the maximal monotone and some convergence problem for strictly pseudocontractive mappings. Our results include the previous results as special cases extend and improve the results of Ceng et al., [Math. Meth. Oper. Res., 67:375–390, 2008] and many others.

The split common fixed point problem is an inverse problem that consists in finding an element in a fixed point set such that its image under a bounded linear operator belongs to another fixed-point set. In this paper, we present new iterative algorithms for solving the split common fixed point problem of demimetric mappings in Hilbert spaces. Moreover, our algorithm does not need any prior information of the operator norm. Weak and strong convergence theorems are given under some mild assumptions. The results in this paper are the extension and improvement of the recent results in the literature.

ABSTRACTIn this paper, we consider the split common fixed point problem for new demimetric mappings in two Banach spaces. Using the hybrid method, we prove a strong convergence theorem for finding a solution of the split common fixed point problem in two Banach spaces. Furthermore, using the shrinking projection method, we obtain another strong convergence theorem for finding a solution of the problem in two Banach spaces. Using these results, we obtain well-known and new strong convergence theorems in Hilbert spaces and Banach spaces.

This paper deals with a variational inequality problem over a solution set of another variational inequality problem, which essentially is called the double-hierarchical constrained optimization problem. Inertial-based algorithms are devised for this problem and solution theorems of strong convergence are established. Simulation results on numerical examples are given to illustrate the behaviour of our proposed algorithms compared with some existing ones.

This paper introduces a novel iterative scheme that integrates an inertial technique and Nesterov acceleration for solving the multi-output set split common fixed point problem with k-semi-contractive mappings. The primary objectives are to enhance the convergence rate and reduce computational complexity. Firstly, under the condition of fixed step size, two iterative algorithms are designed, and their weak convergence and strong convergence are proved respectively. The strong convergence result is achieved by introducing relaxation parameters and external control sequences. Secondly, to avoid the difficulty of operator norm calculation, a variable step size iterative scheme is further proposed, which improves the practicality of the algorithm by dynamically adjusting the step size, and its convergence is strictly analyzed. The theoretical proof shows that under the conditions of satisfying the subclosed principle and appropriate parameters, the sequence generated by the proposed algorithm can converge to the solution of the problem. Numerical experiments verify the acceleration effect of the inertial term and the advantages of variable step size.