An extragradient approximation method for variational inequality problem on fixed point problem of nonexpensive mappings and monotone mappings

Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization

We apply 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.

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

We introduce a new iterative scheme for finding a common element of the set of solutions of a general system of variational inequalities, the set of solutions of a mixed equilibrium problem, and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Using the demiclosedness principle for nonexpansive mappings, we prove that the iterative sequence converges strongly to a common element of the above three sets under some control conditions, and we also give some examples for mappings which satisfy conditions of the main result.

In the literature, various iterative methods have been proposed for finding a common solution of the classical variational inequality problem and a fixed point problem. Research along these lines is performed either by relaxing the assumptions on the mappings in the settings (for instance, commonly seen assumptions for the mapping involved in the fixed point problem are nonexpansive or strictly pseudocontractive) or by adding a general system of variational inequalities into the settings. In this paper, we consider both possible ways in our settings. Specifically, we propose an iterative method for finding a common solution of the classical variational inequality problem, a general system of variational inequalities and a fixed point problem of a uniformly continuous asymptotically strictly pseudocontractive mapping in the intermediate sense. Our iterative method is hybridized by utilizing the well-known extragradient method, the CQ method, the Mann-type iterative method and the viscosity approximation method. The iterates yielded by our method converge strongly to a common solution of these three problems. In addition, we propose a hybridized extragradient-like method to yield iterates converging weakly to a common solution of these three problems. MSC:49J30, 47H09, 47J20.

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.

We consider a new class of multiplier interior point methods for solving variational inequality problems with maximal monotone operators and explicit convex constraint inequalities. Developing a simple Lagrangian duality scheme which is combined with the recent logarithmic-quadratic proximal (LQP) theory introduced by the authors, we derive three algorithms for solving the variational inequality (VI) problem. This provides a natural extension of the methods of multipliers used in convex optimization and leads to smooth interior point multiplier algorithms. We prove primal, dual, and primal-dual convergence under very mild assumptions, eliminating all the usual assumptions used until now in the literature for related algorithms. Applications to complementarity problems are also discussed.

In a real Hilbert space, let GSVI and CFPP represent a general system of variational inequalities and a common fixed point problem of a countable family of nonexpansive mappings and an asymptotically nonexpansive mapping, respectively. In this paper, via a new subgradient extragradient implicit rule, we introduce and analyze two iterative algorithms for solving the monotone bilevel equilibrium problem (MBEP) with the GSVI and CFPP constraints, i.e., a strongly monotone equilibrium problem over the common solution set of another monotone equilibrium problem, the GSVI and the CFPP. Some strong convergence results for the proposed algorithms are established under the mild assumptions, and they are also applied for finding a common solution of the GSVI, VIP, and FPP, where the VIP and FPP stand for a variational inequality problem and a fixed point problem, respectively.

The alternating direction method is an attractive method for solving large-scale variational inequality problems whenever the subproblems can be solved efficiently. However, the subproblems are still variational inequality problems, which are as structurally difficult to solve as the original one. To overcome this disadvantage, in this paper we propose a new alternating direction method for solving a class of nonlinear monotone variational inequality problems. In each iteration the method just makes an orthogonal projection to a simple set and some function evaluations. We report some preliminary computational results to illustrate the efficiency of the method.

In this work, we concern ourselves with the problem of solving a general system of variational inequalities whose solutions also solve a common fixed-point problem of a family of countably many nonlinear operators via a hybrid viscosity implicit iteration method in 2 uniformly smooth and uniformly convex Banach spaces. An application to common fixed-point problems of asymptotically nonexpansive and pseudocontractive mappings and variational inequality problems for strict pseudocontractive mappings is also given in Banach spaces.

In this present paper, we propose a modified form of generalized system of variational inequalities and introduce an iterative scheme for finding a common element of the set of fixed points of nonexpansive mapping and the solution set of the proposed problem in the framework of real Hilbert spaces. We prove a strong convergence theorem of the proposed iterative scheme. Applying our main result, we prove strong convergence theorems of the standard constrained convex optimization problem and the split feasibility problem. In support of our main result, a numerical example is also presented.

<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>

This paper presents an enhanced inertial Tseng’s extragradient method designed to address variational inequality problems involving pseudomonotone operators, along with fixed point problems governed by quasi-nonexpansive operators in real Hilbert spaces. Provided that the parameters satisfy appropriate conditions, the proposed method is shown to converge strongly. Finally, we provide computational results and illustrate their utility through optimal control applications. These aim to show the efficacy and superiority of the proposed algorithm compared with some existing algorithms.

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 introduce a modified Tseng's extragradient method with a new step-length rule to solve pseudo-monotone variational inequalities in real Hilbert spaces. Under suitable conditions, the sequence generated by this algorithm strongly converges to the common elements of the solution set of pseudo-monotone variational inequality problems and the fixed point set of k-demicontractive mappings. Finally, we give some numerical experiments to illustrate the effectiveness of the proposed algorithm. The main results of this paper generalize and improve some known results in the literature.