Hierarchical Variational Inequality Research Articles

Composite relaxed extragradient method for triple hierarchical variational inequalities with constraints of systems of variational inequalities

Composite relaxed extragradient method for triple hierarchical variational inequalities with constraints of systems of variational inequalities

Hybrid steepest-descent viscosity methods for triple hierarchical variational inequalities with constraints of mixed equilibria and bilevel variational inequalities

Hybrid steepest-descent viscosity methods for triple hierarchical variational inequalities with constraints of mixed equilibria and bilevel variational inequalities

Multi-step hybrid steepest-descent methods for split feasibility problems with hierarchical variational inequality problem constraints

Multi-step hybrid steepest-descent methods for split feasibility problems with hierarchical variational inequality problem constraints

Some Existence and Convergence Theorems for Solving a System of Hierarchical Optimization Problems

In this article, we prove the existence and convergence theorems of the solution for a system of hierarchical variational inequality problem in Hilbert spaces. In this paper, we use Mainge's approach for finding a solution of the system of hierarchical variational inequality problems. Our result in this article improves, extends, and generalizes some well-known corresponding results in the literature.

Hybrid extragradient method with regularization for triple hierarchical variational inequalities with general mixed equilibrium and split feasibility constraints

In this paper, we introduce a hybrid extragradient iterative algorithm with regularization for solving the triple hierarchical variational inequality problem (THVIP) (defined over the common fixed point set of finitely many nonexpansive mappings and a strictly pseudocontraction) with constraints of a general mixed equilibrium problem (GMEP), a split feasibility problem (SFP) and a general system of variational inequalities (GSVI). The iterative algorithm is based on Korpelevich's extragradient method, viscosity approximation method, Mann's iteration method, hybrid steepest descent method and gradient-projection method (GPM) with regularization. It is proven that, under very mild conditions, the sequences generated by the proposed algorithm converge strongly to a unique solution of the THVIP. We also give the applications of our results for solving some special cases of the THVIP. The results presented in this paper improve and extend some corresponding ones in the earlier and recent literature.

Split hierarchical variational inequality problems and fixed point problems for nonexpansive mappings

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.

Mann-type hybrid steepest-descent method for three nonlinear problems

We introduce a Mann-type hybrid steepest-descent iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of solutions of finitely many variational inequalities, and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical variational inequality.

An iterative method for split hierarchical monotone variational inclusions

AbstractIn this paper, we introduce a split hierarchical monotone variational inclusion problem (SHMVIP) which includes split variational inequality problems, split monotone variational inclusion problems, split hierarchical variational inequality problems, etc., as special cases. An iterative algorithm is proposed to compute the approximate solutions of an SHMVIP. The weak convergence of the sequence generated by the proposed algorithm is studied. We present an example to illustrate our algorithm and convergence result.

Optimal competitive freight network design as hierarchical variational inequalities programming problems

Freight networks are a case of systems that multiple participants are composing interrelations along the complete supply chain. Their interrelations correspond to alternative behavior, namely, cooperation, non-cooperation and competition, while they are large-scale spatially distributed systems combining multiple means of transportation and the infrastructure and equipment typically utilized for servicing demand, results to a complex system integration. In this paper, the case of the optimal design of freight networks is investigated, aiming to highlight the particularities emerging in this case of transportation facilities strategic and/or operational planning and the multiple game-theoretic and equilibrium problems that are structured in cascade and in hierarchies. The application that is investigated here focuses in the design of a significant ‘player’ of the freight supply chain, namely container terminals, while the proposed framework will aim on analyzing investment strategies built on integrated demand–supply models and the optimal network design format. The approach will build on the multilevel Mathematical Programming with Equilibrium Constraints (MPECs) formulation, but is further extended to cope with the properties introduced by the ‘designers’ (infrastructure authorities), shippers and carriers competition in all levels of MPECs. Since container terminals are typically competing each other, the nomenclature used here for formulating appropriate MPECs problems are based on hierarchies of Variational Inequalities (VI) problems, able to capture the alternative relationships emerging in realistic freight supply chains. The proposed formulations of the competitive network design case is addressed by a novel approach of co-evolutionary agents, which can be regarded as new in equilibrium estimation. Finally, the results are compared with alternative network design cases, namely the centralized cooperative and exchanging design. Under this analysis it is able to highlight the differences among alternative design cases, but moreover an estimation of the ‘price of anarchy’ in transportation systems design is offered, an element of both theoretical as well as practical relevance.

Hybrid extragradient viscosity method for general system of variational inequalities

In this paper, we introduce a hybrid extragradient viscosity iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of a general system of variational inequalities, the set of solutions of a split feasibility problem (SFP), and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. The iterative algorithm is based on Korpelevich’s extragradient method, viscosity approximation method, Mann’s iteration method, hybrid steepest-descent method and gradient-projection method (GPM) with regularization. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical variational inequality.

Iterative methods for triple hierarchical variational inequalities with mixed equilibrium problems, variational inclusions, and variational inequalities constraints

In this paper, we introduce and analyze a multi-step hybrid steepest-descent extragradient algorithm and multi-step composite Mann-type viscosity iterative algorithm for finding a solution of triple hierarchical variational inequalities defined over the common set of solutions of mixed equilibrium problems, variational inclusions, variational inequalities, and fixed point problems. Under appropriate assumptions, we prove that the proposed algorithms converge strongly to a common element of the fixed point set of a strict pseudocontractive mapping, a solution set of finitely many generalized mixed equilibrium problems, a solution set of finitely many variational inclusions, and a solution set of a general system of variational inequalities. Such an element is a unique solution of a triple hierarchical variational inequality problem. In addition, we also consider as an application the proposed algorithm to solve a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, and a general system of variational inequalities. The results obtained in this paper improve and extend the corresponding results announced by many other authors.

Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

We introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove 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 inclusions, and the solution set of general system of variational inequalities (GSVI), 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 for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and the GSVI. The results obtained in this paper improve and extend the corresponding results announced by many others.

Multi-step extragradient method with regularization for triple hierarchical variational inequalities with variational inclusion and split feasibility constraints

AbstractBy combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, Mann’s iteration method, and the gradient-projection method with regularization, a hybrid multi-step extragradient algorithm with regularization for finding a solution of triple hierarchical variational inequality problem is introduced and analyzed. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a unique solution of a triple hierarchical variational inequality problem which is defined over the set of solutions of a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems (GMEP), finitely many variational inclusions, fixed point problems, and the split feasibility problem (SFP). We also prove the strong convergence of the proposed algorithm to a common solution of the SFP, finitely many GMEPs, finitely many variational inclusions, and the fixed point problem of a strict pseudocontraction. The results presented in this paper improve and extend the corresponding results announced by several others.MSC:49J30, 47H09, 47J20, 49M05.

Regularized hybrid iterative algorithms for triple hierarchical variational inequalities

AbstractIn this paper, we introduce and study a triple hierarchical variational inequality (THVI) with constraints of minimization and equilibrium problems. More precisely, letFix(T)be the fixed point set of a nonexpansive mapping, letMEP(Θ,φ)be the solution set of a mixed equilibrium problem (MEP), and letΓbe the solution set of a minimization problem (MP) for a convex and continuously Frechet differential functional in Hilbert spaces. We want to find a solutionx∗∈Fix(T)∩MEP(Θ,φ)∩Γof a variational inequality with a variational inequality constraint over the intersection ofFix(T),MEP(Θ,φ), andΓ. We propose a hybrid iterative algorithm with regularization to compute approximate solutions of the THVI, and we present the convergence analysis of the proposed iterative algorithm.MSC:49J40, 47J20, 47H10, 65K05, 47H09.

Iterative methods for triple hierarchical variational inequalities and common fixed point problems

AbstractThe purpose of this paper is to introduce a new iterative scheme for approximating the solution of a triple hierarchical variational inequality problem. Under some requirements on parameters, we study the convergence analysis of the proposed iterative scheme for the considered triple hierarchical variational inequality problem which is defined over the set of solutions of a variational inequality problem defined over the intersection of the set of common fixed points of a sequence of nearly nonexpansive mappings and the set of solutions of the classical variational inequality. Our strong convergence theorems extend and improve some known corresponding results in the contemporary literature for a wider class of nonexpansive type mappings in Hilbert spaces.MSC:47J20, 47J25.

A unified extragradient method for systems of hierarchical variational inequalities in a Hilbert space

Abstract In this paper, we introduce and analyze a multistep Mann-type extragradient iterative algorithm by combining Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method, Mann’s iteration method, and the projection method. 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 and a strict pseudocontraction, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inclusions and the solution set of a variational inequality problem (VIP), which is just a unique solution of a system of hierarchical variational inequalities (SHVI) in a real Hilbert space. The results obtained in this paper improve and extend the corresponding results announced by many others. MSC:49J30, 47H09, 47J20, 49M05.

Extragradient method for convex minimization problem

Abstract In this paper, we introduce and analyze a multi-step hybrid extragradient algorithm by combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, Mann’s iteration method and the gradient-projection method (GPM) with regularization in the setting of infinite-dimensional Hilbert spaces. It is proven that, under appropriate assumptions, the proposed algorithm converges strongly to a solution of the convex minimization problem (CMP) with constraints of several problems: finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to the unique solution of a hierarchical variational inequality problem (over the fixed point set of a strictly pseudocontractive mapping) with constraints of finitely many GMEPs, finitely many variational inclusions and the CMP. The results presented in this paper improve and extend the corresponding results announced by many others. MSC:49J30, 47H09, 47J20, 49M05.

Hybrid extragradient method for hierarchical variational inequalities

AbstractIn this paper, we consider a hierarchical variational inequality problem (HVIP) defined over a common set of solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, a general system of variational inequalities, and the fixed point problem of a strictly pseudocontractive mapping. By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method and Mann’s iteration method, we introduce and analyze a multistep hybrid extragradient algorithm for finding a solution of our HVIP. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a solution of a general system of variational inequalities defined over a common set of solutions of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to a unique solution of our HVIP. The results obtained in this paper improve and extend the corresponding results announced by many others.MSC:49J30, 47H09, 47J20.

Multi-step iterative algorithms with regularization for triple hierarchical variational inequalities with constraints of mixed equilibria, variational inclusions, and convex minimization

Abstract In this paper, we introduce and analyze a relaxed iterative algorithm by virtue of Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, the regularization method, and the 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 inclusions 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 solving a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many variational inclusions, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others. MSC:49J30, 47H09, 47J20, 49M05.

Split hierarchical variational inequality problems and related problems

Abstract The main objective of this paper is to introduce a split hierarchical variational inequality problem. Several related problems are also considered. We propose an iterative method for finding a solution of our problem. The weak convergence of the sequence generated by the proposed method is studied. MSC:47H09, 47H10, 47J25, 49J40, 65K10.