Quasi-Strict Feasibility of Generalized Mixed Variational Inequalities in Reflexive Banach Spaces

In this paper, quasi-strict feasibility of a generalized mixed variational inequality as a new notation is introduced, which is weaker than its strict feasibility and recovers the existing concept of strict feasibility for a generalized variational inequality. By using the equivalent characterization of the nonemptiness and boundedness of the solution set for the generalized mixed variational inequality, it is proved that quasi-strict feasibility is a sufficient condition for the generalized mixed variational inequality with a f-pseudomonotone and upper hemicontinuous mapping to have a nonempty and bounded solution set in reflexive Banach spaces. Our results generalize and extend some known results in Zhong and Huang (J Optim Theory Appl 152(3):696–709, 2012).     

Strict feasibility and solvability for vector equilibrium problems in reflexive Banach spaces

The purpose of this paper is to investigate nonemptiness and boundedness of the solution set for a vector equilibrium problem with strict feasibility in reflexive Banach spaces. We introduce the concept of strict feasibility for a vector equilibrium problem, which recovers the existing concepts of strict feasibility introduced for variational inequalities. We prove that a pseudomonotone vector equilibrium problem has a nonempty bounded solution provided that it is strictly feasible in the strong sense.     

Characterizations of Levitin–Polyak well-posedness by perturbations for the split variational inequality problem

The purpose of this paper is to investigate Levitin–polyak well-posedness by perturbations of the split variational inequality problem in reflexive Banach spaces. Furi-Vignoli-type characterizations are established for the well-posedness. We prove that the weak generalized Levitin–Polyak well-posedness by perturbations is equivalent to the nonemptiness and boundedness of the solution set of the problem. Finally, we discuss the relations between the Levitin–Polyak well-posedness by perturbations of the split variational inequality problem and the Levitin–Polyak well-posedness by perturbations of the split minimization problem when the split variational inequality problem arises from the split minimization problem.     

Stability Analysis for Minty Mixed Variational Inequality in Reflexive Banach Spaces

This paper is devoted to the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. Several equivalent characterizations are given for the Minty mixed variational inequality to have nonempty and bounded solution set. A stability result is presented for the Minty mixed variational inequality with Φ-pseudomonotone mapping in reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. As an application, a stability result for a generalized mixed variational inequality is also obtained. The results presented in this paper generalize and extend some known results in Fan and Zhong (Nonlinear Anal., Theory Methods Appl. 69:2566–2574, 2008) and He (J. Math. Anal. Appl. 330:352–363, 2007).     

Existence results for solutions of mixed tensor variational inequalities

By employing the notion of exceptional family of elements, we establish existence results for the mixed tensor variational inequalities. We show that the nonexistence of an exceptional family of elements is a sufficient condition for the solvability of mixed tensor variational inequality. For positive semidefinite mixed tensor variational inequalities, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the nonemptiness of the solution sets. We derive several sufficient conditions of the nonemptiness and compactness of the solution sets for the mixed tensor variational inequalities with some special structured tensors. Finally, we show that the mixed tensor variational inequalities can be defined as a class of convex optimization problems.

     

Existence results for a class of hemivariational inequality problems on Hadamard manifolds

In this paper, a class of hemivariational inequality problems are introduced and studied on Hadamard manifolds. Using the properties of Clarke’s generalized directional derivative and Fan-KKM lemma, an existence theorem of solution in connection with the hemivariational inequality problem is obtained when the constraint set is bounded. By employing some coercivity conditions and the properties of Clarke’s generalized directional derivative, an existence result and the boundedness of the set of solutions for the underlying problem are investigated when the constraint set is unbounded. Moreover, a sufficient and necessary condition for ensuring the nonemptiness of the set of solutions concerned with the hemivariational inequality problem is also given.     

Strict feasibility and stable solvability of bifunction variational inequalities

In this paper, we study strict feasibility of a bifunction variational inequality. It is proved that a monotone bifunction variational inequality has a nonempty and bounded solution set if and only if it is strictly feasible. Stable solvability of the bifunction variational inequality is discussed under strict feasibility assumption when the domain set is perturbed. Our results generalize earlier results on the classical variational inequality to the case of the bifunction variational inequality.     

On the nonemptiness and compactness of the solution sets for vector variational inequalities

In this article, we investigate conditions for nonemptiness and compactness of the sets of solutions of pseudomonotone vector variational inequalities by using the concept of asymptotical cones. We show that a pseudomonotone vector variational inequality has a nonempty and compact solution set provided that it is strictly feasible. We also obtain some necessary conditions for the set of solutions of a pseudomonotone vector variational inequality to be nonempty and compact.     

Characterizing the Nonemptiness and Compactness of the Solution Set of a Vector Variational Inequality by Scalarization

In this paper, the nonemptiness and compactness of the solution set of a pseudomonotone vector variational inequality defined in a finite-dimensional space are characterized in terms of that of the solution sets of a family of linearly scalarized variational inequalities.     

Outer approximation schemes for generalized semi-infinite variational inequality problems

We introduce and analyse outer approximation schemes for solving variational inequality problems in which the constraint set is as in generalized semi-infinite programming. We call these problems generalized semi-infinite variational inequality problems. First, we establish convergence results of our method under standard boundedness assumptions. Second, we use suitable Tikhonov-like regularizations for establishing convergence in the unbounded case.     

Gap Functions for Quasivariational Inequalities and Generalized Nash Equilibrium Problems

The gap function (or merit function) is a classic tool for reformulating a Stampacchia variational inequality as an optimization problem. In this paper, we adapt this technique for quasivariational inequalities, that is, variational inequalities in which the constraint set depends on the current point. Following Fukushima (J. Ind. Manag. Optim. 3:165–171, 2007), an axiomatic approach is proposed. Error bounds for quasivariational inequalities are provided and an application to generalized Nash equilibrium problems is also considered.     

Characterizations of the Nonemptiness and?Boundedness of Weakly Efficient Solution Sets of?Convex Vector Optimization Problems in Real Reflexive?Banach Spaces

Under a weak compactness assumption on the functions involved, which always holds in finite-dimensional normed linear spaces, this paper extends various characterizations of the nonemptiness and boundedness of weakly efficient solution sets of convex vector optimization problems, obtained previously by the author (Deng in J. Optim. Theory Appl. 96:123–131, 1998) in the real finite-dimensional normed linear space setting, to those in the real reflexive Banach space setting.     

Generalized equilibrium problems and fixed point problems for nonexpansive semigroups in Hilbert spaces

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of solutions of the generalized equilibrium problems and the set of all common fixed points of a nonexpansive semigroup in the framework of a real Hilbert space. We prove that both approaches converge strongly to a common element of such two sets. Such common element is the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Furthermore, we utilize the main results to obtain two mean ergodic theorems for nonexpansive mappings in a Hilbert space. The results of this paper extend and improve the results of Li et al. (J Nonlinear Anal 70:3065–3071, 2009), Cianciaruso et al. (J Optim Theory Appl 146:491–509, 2010) and many others.     

Merit functions and error bounds for constrained mixed set-valued variational inequalities via generalized f-projection operators

In this paper, we introduce and investigate a constrained mixed set-valued variational inequality (MSVI) in Hilbert spaces. We prove the solution set of the constrained MSVI is a singleton under strict monotonicity. We also propose four merit functions for the constrained MSVI, that is, the natural residual, gap function, regularized gap function and D-gap function. We further use these functions to obtain error bounds, i.e. upper estimates for the distance to solutions of the constrained MSVI under strong monotonicity and Lipschitz continuity. The approach exploited in this paper is based on the generalized f-projection operator due to Wu and Huang, but not the well-known proximal mapping.     

Regularization of quasi-variational inequalities

An ill-posed quasi-variational inequality with contaminated data can be stabilized by employing the elliptic regularization. Under suitable conditions, a sequence of bounded regularized solutions converges strongly to a solution of the original quasi-variational inequality. Moreover, the conditions that ensure the boundedness of regularized solutions, become sufficient solvability conditions. It turns out that the regularization theory is quite strong for quasi-variational inequalities with set-valued monotone maps but restrictive for generalized pseudo-monotone maps. The results are quite general and are applicable to ill-posed variational inequalities, hemi-variational inequalities, inverse problems, and split feasibility problem, among others.     

Localization of Generalized Normal Maps and Stability of Variational Inequalities in Reflexive Banach Spaces

In this paper, we introduce a localized version of generalized normal maps as well as generalized natural mappings. By using these concepts, we study continuity properties of the solution map of parametric variational inequalities in reflexive Banach spaces. This localization permits us to deal with variational conditions posed on sets that may not be convex and to establish existence and continuity of solutions. We also establish homeomorhisms between the solution set of variational inequalities and the solution set of generalized normal maps. Using these homeomorphisms and the degree theory, we show that the solution map of parametric variational inequalities is lower semicontinuous. Our results extend some results of Robinson (Set-Valued Anal 12:259–274, 2004). The authors wish to express their sincere appreciation to Professor Stephen M. Robinson, Department of Industrial and Systems Engineering, University of Wisconsin-Madison, for his valuable comments and suggestions. This research was partially supported by a grant from National Science Council of Taiwan, ROC.     

A Class of Generalized Evolutionary Problems Driven by Variational Inequalities and Fractional Operators

This paper is devoted to a generalized evolution system called fractional partial differential variational inequality which consists of a mixed quasi-variational inequality combined with a fractional partial differential equation in a Banach space. Invoking the pseudomonotonicity of multivalued operators and a generalization of the Knaster-Kuratowski-Mazurkiewicz theorem, first, we prove that the solution set of the mixed quasi-variational inequality involved in system is nonempty, closed and convex. Next, the measurability and upper semicontinuity for the mixed quasi-variational inequality with respect to the time variable and state variable are established. Finally, the existence of mild solutions for the system is delivered. The approach is based on the theory of operator semigroups, the Bohnenblust-Karlin fixed point principle for multivalued mappings, and theory of fractional operators.

     

Stable pseudomonotone variational inequality in reflexive Banach spaces

Stability of a generalized variational inequality with either the mapping or the set perturbed is discussed in reflexive Banach spaces, provided that the mappings are pseudomonotone in the sense of Karamardian. As a byproduct, generalized variational inequality having nonempty and bounded set is proved to be equivalent to the strictly feasibility.     

非强制混合向量变分不等式解的存在性研究

本文利用例外簇方法研究非强制混合向量变分不等式的弱有效解的存在性:首先证明若混合向量变分不等式问题不存在例外簇,则混合向量变分不等式问题的弱有效解集为非空集合:利用向量值映射的渐近映射给出自反Banach空间中非强制混合向量变分不等式的弱有效解集不存在例外簇的充分条件,从而得到混合向量变分不等式问题的弱有效解的存在性结果;我们研究了当算子为余正仿射算子时,给出混合仿射向量变分不等式不存在例外簇的充分条件,得到混合仿射向量变分不等式弱有效解的存在性,给出了混合仿射向量变分不等式的弱有效解集为非空紧致集的充分条件.将Iusem等人(2019)在有限维空间中标量混合变分不等式解的存在性结果推广到自反Banach空间中混合向量变分不等式.