Set-valued Variational Inequalities Research Articles

Set-valued equilibrium problems based on the concept of null set with applications

In this paper, by means of the null set defined by Wu (J Math Anal Appl. 2019;472:1741–1761), we introduce a set-valued equilibrium problem based on the null set, where the objective mapping takes values in a hyperspace equipped with a convex cone. Moreover, we obtain new existence results for set-valued equilibrium problems defined on compact or noncompact sets. Some applications are given to set optimization problems, to a set-valued variational inequality, to saddle point theorems for set-valued mappings, and to a generalized noncooperative game.

An extragradient method for non-monotone equilibrium problems on Hadamard manifolds with applications

In the present paper, we consider a non-monotone equilibrium problem on Hadamard manifolds and define Armijo's type extragradient algorithm for the proposed equilibrium problem. The introduced algorithm does not require objective bifunction's monotonicity and the solution set's nonemptiness. A convergence result of our algorithm under very mild assumptions is presented. Moreover, we investigate the applications of the established results to the non-monotone set-valued variational inequalities and the generalized Nash equilibrium problems from the considered method.

Brezis pseudomonotone bifunctions and quasi equilibrium problems via penalization

In this paper we investigate quasi equilibrium problems in a real Banach space under the assumption of Brezis pseudomonotonicity of the function involved. To establish existence results under weak coercivity conditions we replace the quasi equilibrium problem with a sequence of penalized usual equilibrium problems. To deal with the non compact framework, we apply a regularized version of the penalty method. The particular case of set-valued quasi variational inequalities is also considered.

Auxiliary Principle Technique for Hierarchical Equilibrium Problems

In this paper, building upon auxiliary principle technique and using proximal operator, we introduce a new explicit algorithm for solving monotone hierarchical equilibrium problems. The considered problem is a monotone equilibrium problem, where the constraint is the solution set of a set-valued variational inequality problem. The strong convergence of the proposed algorithm is studied under strongly monotone and Lipschitz-type assumptions of the bifunction. By combining with parallel techniques, the convergence result is also established for the equilibrium problem involving a finite system of demicontractive mappings. Several fundamental experiments are provided to illustrate the numerical behavior of the proposed algorithm and comparison with other known algorithms is studied.

Convergence of Some Iterative Algorithms for System of Generalized Set-Valued Variational Inequalities

In this article, we consider and study a system of generalized set-valued variational inequalities involving relaxed cocoercive mappings in Hilbert spaces. Using the projection method and Banach contraction principle, we prove the existence of a solution for the considered problem. Further, we propose an iterative algorithm and discuss its convergence. Moreover, we establish equivalence between the system of variational inequalities and altering points problem. Some parallel iterative algorithms are proposed, and the strong convergence of the sequences generated by these iterative algorithms is discussed. Finally, a numerical example is constructed to illustrate the convergence analysis of the proposed parallel iterative algorithms.

Perturbation strategy for splitting operator method to solve the set-valued variational inequalities

Perturbation strategy for splitting operator method to solve the set-valued variational inequalities

Regularization of Brézis pseudomonotone variational inequalities

In this paper we prove the existence of solutions of regularized set-valued variational inequalities involving Brezis pseudomonotone operators in reflexive and locally uniformly convex Banach spaces. By taking advantage of this result, we approximate a general set-valued variational inequality with suitable regularized set-valued variational inequalities, and we show that their solutions weakly converge to a solution of the original one. Furthermore, by strengthening the coercivity conditions, we can prove the strong convergence of the approximate solutions.

Globally proper efficiency of set-valued optimization and vector variational inequality involving the generalized contingent epiderivative

In this paper, firstly, a new property of the cone subpreinvex set-valued map involving the generalized contingent epiderivative is obtained. As an application of this property, a sufficient optimality condition for constrained set-valued optimization problem in the sense of globally proper efficiency is derived. Finally, we establish the relations between the globally proper efficiency of the set-valued optimization problem and the globally proper efficiency of the vector variational inequality.

Limit vector variational inequality problems via scalarization

We solve a general vector variational inequality problem in a finite—dimensional setting, where only approximation sequences are known instead of exact values of the cost mapping and feasible set. We establish a new equivalence property, which enables us to replace each vector variational inequality with a scalar set-valued variational inequality. Then, we approximate the scalar set-valued variational inequality with a sequence of penalized problems, and we study the convergence of their solutions to solutions of the original one.

Auxiliary principle technique for solving regularized nonconvex set-valued mixed variational inequalities

In this paper, we introduce a new class of regularized nonconvex set-valued mixed variational inequalities where the underlying set is uniformly prox-regular. By using the auxiliary principle technique, we propose a predictor-corrector method for solving such class of nonconvex variational inequalities and study the strong convergence of the sequences generated by the proposed algorithm. As a consequence of our main results, we provide the correct version of algorithms and results presented in the literature.

Weak minimizers, minimizers and variational inequalities for set-valued functions. A blooming wreath?

Recently, necessary and sufficient conditions in terms of variational inequalities have been introduced to characterize minimizers of convex set-valued functions. Similar results have been proved for a weaker concept of minimizers and weaker variational inequalities. The implications are proved using scalarization techniques that eventually provide original problems, not fully equivalent to the set-valued counterparts. Therefore, we try, in the course of this note, to close the network among the various notions proposed. More specifically, we prove that a minimizer is always a weak minimizer, and a solution to the stronger variational inequality always also a solution to the weak variational inequality of the same type. As a special case, we obtain a complete characterization of efficiency and weak efficiency in vector optimization by set-valued variational inequalities and their scalarizations. Indeed, this might eventually prove the usefulness of the set optimization approach to renew the study of vector optimization.

A Projection-Type Method for Set Valued Variational Inequality Problems on Hadamard Manifolds

In this paper, we develop a projection-type algorithm for set-valued variational inequalities on Hadamard manifolds. The proposed method is well defined whether the solution set of the problem is non-empty or not. Under pseudomonotonicity assumptions on the underlying vector field, our method is convergent to a solution of the given set-valued variational inequality. The results presented in this paper generalize and improve some known results introduced by Tang et al. (Optimization 64(5):1081–1096, 2015).

Weak sharpness for set-valued variational inequalities and applications to finite termination of iterative algorithms

In this paper, we introduce the notion of weak sharpness for set-valued variational inequalities in the n-dimensional Euclidean space and then present some characterizations of weak sharpness. We also give some examples to illustrate this notion. Under the assumption of weak sharpness, by using the inner limit of a set sequence we establish a sufficient and necessary condition to guarantee the finite termination of an arbitrary algorithm for solving a set-valued variational inequality involving maximal monotone mappings. As an application, we show that the sequence generated by the hybrid projection-proximal point algorithm proposed by Solodov and Svaiter terminates at solutions in a finite number of iterations. These obtained results extend some known results of classical variational inequalities.

Gap Functions and Error Bounds for Generalized Set-Valued Mixed Variational Inequality

In this paper, we introduce some gap functions for generalized set-valued mixed variational inequalities to obtain the equivalent optimization problems for these generalized set-valued mixed variational inequalities. We use these gap functions to obtain error bounds for solutions of generalized set-valued mixed variational inequalities in Hilbert spaces. The results obtained in this paper generalize some corresponding known results in literature.

Approximate solutions for nonconvex set-valued optimization and vector variational inequality

This note deals with approximate solutions in vector optimization involving a generalized cone-invex set-valued mapping. First, a new class of generalized cone-invex set-valued maps, called cone-subinvex set-valued maps, is introduced. Then the sufficient optimality condition and two types dual theorems are established for weakly approximate minimizers under the assumption of cone-subinvexity. Finally, it also reveals the closed relationships between a weakly approximate minimizer of a cone-subinvex set-valued optimization problem and a weakly approximate solution of a kind of vector variational inequality.

Shares Allocation Methods for Generalized Game Problems with Joint Constraints

We consider an extension of a noncooperative game problem where players have joint binding constraints. We suggest a shares allocation approach, which replaces the initial problem with a sequence of Nash equilibrium problems together with an upper level set-valued variational inequality as master problem. This transformation maintains the monotonicity properties of the underlying mappings. We also show that the regularization yields a decomposable penalty method, which removes complex functions in constraints within the custom noncooperative game framework and provides the single-valued master problem with strengthened monotonicity of its cost mapping.

Degree Theory and Solution Existence of Set-Valued Vector Variational Inequalities in Reflexive Banach Spaces

In this paper, a degree theory for set-valued vector variational inequalities is built in reflexive Banach spaces. By using the method of degree theory, some existence results of solutions for set-valued vector variational inequalities are established under suitable conditions. Furthermore, some equivalent characterizations for the nonemptiness and boundedness of solution sets to single-valued vector variational inequalities are obtained under pseudomonotonicity assumption. To the best of our knowledge, there are still no papers dealing with the degree theory for vector variational inequalities.

A projection algorithm for set-valued variational inequalities on Hadamard manifolds

A projection algorithm is suggested for the set-valued variational inequality on Hadamard manifold. Under a pseudomonotone assumption on the underlying vector field, our method is proved to be globally convergent to a solution of variational inequality. This algorithm is a natural extension to the Hadamard manifold of the one introduced by Fang and He (Appl Math Comput, 217:9543–9511, 2011) in $${\mathbb {R}}^n$$ .

Some generalizations of common fixed point problems with applications

In this paper, we establish a unified approach to study the existence of fixed points for common fixed point problems. Moreover, we introduce mixed systems of common fixed point problems, systems of common fixed point problems and common fixed point problems without convex assumptions. As applications, we establish a general type of some set-valued variational inequalities, and we obtain some existence theorems of solutions of set-valued variational inequalities. The results of this paper improve and generalize several known results on common fixed point problems.

An Inexact Penalty Method for Non Stationary Generalized Variational Inequalities

We consider a set-valued (generalized) variational inequality problem in a finite-dimensional setting, where only approximation sequences are known instead of exact values of the cost mapping and feasible set. We suggest to apply a sequence of inexact solutions of auxiliary problems involving general penalty functions. Its convergence is attained without concordance of penalty, accuracy, and approximation parameters under certain coercivity type conditions.