On the Stability of Generalized Vector Quasivariational Inequality Problems

In this paper, we obtain some stability results for generalized vector quasivariational inequality problems. We prove that the solution set is a closed set and establish the upper semicontinuity property of the solution set for perturbed generalized vector quasivariational inequality problems. These results extend those obtained in Ref. 1. We obtain also the lower semicontinuity property of the solution set for perturbed classical variational inequalities. Several examples are given for the illustration of our results.     

On the existence of solutions to generalized quasi-equilibrium problems

In this paper, we apply new results on variational relation problems obtained by D. T. Luc (J Optim Theory Appl 138:65–76, 2008) to generalized quasi-equilibrium problems. Some sufficient conditions on the existence of its solutions of generalized quasi-equilibrium problems are shown. As special cases, we obtain several results on the existence of solutions of generalized Pareto and weak quasi-equilibrium problems concerning C-pseudomonotone multivalued mappings. We deduce also some results on the existence of solutions to generalized vector Pareto and weakly quasivariational inequality and vector Pareto quasi-optimization problems with multivalued mappings.     

On the Existence of Solutions to Vector Quasivariational Inequalities and Quasicomplementarity Problems with Applications Break to Traffic Network Equilibria

For vector quasivariational inequalities involving multifunctions in topological vector spaces, an existence result is obtained without a monotonicity assumption and with a convergence assumption weaker than semicontinuity. A new type of quasivariational inequality is proposed. Applications to quasicomplementarity problems and traffic network equilibria are considered. In particular, definitions of weak and strong Wardrop equilibria are introduced for the case of multivalued cost functions.     

On the Existence of Solutions of Quasi-equilibrium Problems with Constraints

The quasi-equilibrium problems with constraints are formulated and some sufficient conditions on the existence of their solutions are shown. As special cases, we obtain several results on the existence of solutions of some vector quasivariational inequality and vector optimization problems. An application of the obtained results is given to show the existence of solutions of quasi-optimization problems with constraints.     

The Solvability of Nonlinear Split Ordered Variational Inequality Problems in Partially Ordered Vector Spaces

The concept of nonlinear split ordered variational inequality problems on partially ordered Banach spaces extends the concept of the linear split vector variational inequality problems on Banach spaces, while the latter is a natural extension of vector variational inequality problems on Banach spaces. In this article, we prove the solvability of some nonlinear split vector variational inequality problems by using fixed-point theorems on partially ordered Banach spaces. It is important to notice that in the results obtained in this article, the considered mappings are not required to have any type of continuity and they just satisfy some order-monotonic conditions. Consequently, both the solvability of linear split vector variational inequality problems and vector variational inequality problems will be immediately obtained from the solvability of nonlinear split vector variational inequality problems. We will apply these results to solving nonlinear split vector optimization problems. The underlying spaces of the considered variational inequality problems may just be vector spaces which do not have topological structures, the considered mappings are not required to satisfy any continuity conditions, which just satisfy some order-increasing conditions.     

Some Existence Results for Vector Quasivariational Inequalities Involving Multifunctions and Applications to Traffic Equilibrium Problems

Some existence results for vector quasivariational inequalities with multifunctions in Banach spaces are derived by employing the KKM-Fan theorem. In particular, we generalize a result by Lin, Yang and Yao, and avoid monotonicity assumptions. We also consider a new quasivariational inequality problem and propose notions of weak and strong equilibria while applying the results to traffic network problems.     

Conjugate duality in vector optimization and some applications to the vector variational inequality

The aim of this paper is to extend the so-called perturbation approach in order to deal with conjugate duality for constrained vector optimization problems. To this end we use two conjugacy notions introduced in the past in the literature in the framework of set-valued optimization. As a particular case we consider a vector variational inequality which we rewrite in the form of a vector optimization problem. The conjugate vector duals introduced in the first part allow us to introduce new gap functions for the vector variational inequality. The properties in the definition of the gap functions are verified by using the weak and strong duality theorems.     

Gap functions and global error bounds for set-valued variational inequalities

The set-valued variational inequality problem is very useful in economics theory and nonsmooth optimization. In this paper, we introduce some gap functions for set-valued variational inequality problems under suitable assumptions. By using these gap functions we derive global error bounds for the solution of the set-valued variational inequality problems. Our results not only generalize the previously known results for classical variational inequalities from single-valued case to set-valued, but also present a way to construct gap functions and derive global error bounds for set-valued variational inequality problems.     

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.     

Levitin–Polyak well-posedness for parametric quasivariational inequality problem of the Minty type

In this paper, we introduce the notions of Levitin?CPolyak (LP) well-posedness and Levitin?CPolyak well-posedness in the generalized sense, for a parametric quasivariational inequality problem of the Minty type. Metric characterizations of LP well-posedness and generalized LP well-posedness, in terms of the approximate solution sets are presented. A parametric gap function for the quasivariational inequality problem is introduced and an equivalence relation between LP well-posedness of the parametric quasivariational inequality problem and that of the related optimization problem is obtained.     

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.     

混合向量变分不等式标量化及间隙函数误差界

利用Konnov对变分不等式问题的标量化方法,对一般的强变分不等式（SVI）和弱变分不等式（WVI）进行了进一步的推广.主要介绍了基于集值映射的强广义混合向量变分不等式（SGMVVI）和弱广义混合向量变分不等式（WGMVVI）,考虑了与它们相关的间隙函数,在合适的条件下讨论了强广义混合集值变分不等式（SGMVI）的间隙函数和SGMVVI的间隙函数之间的关系,以及WGMVVI和SGMVI的间隙函数之间的关系,最后讨论了它们的间隙函数的全局误差界.     

Levitin-Polyak Well-Posedness of Constrained Inverse Quasivariational Inequality

In this article, we introduce and study different types of Levitin–Polyak well-posedness for a constrained inverse quasivariational inequality problem. Criteria and characterizations for these types of well-posedness for inverse quasivariational inequality problems are given. Su?cient conditions for the Levitin–Polyak well-posedness of inverse quasivariational inequality problems are also established.     

On System of Generalized Vector Quasi-equilibrium Problems with Set-valued Maps

In this paper, we introduce four new types of the system of generalized vector quasi-equilibrium problems with set-valued maps which include system of vector quasi-equilibrium problems, system of vector equilibrium problems, system of variational inequality problems, and vector equilibrium problems in the literature as special cases. We prove the existence of solutions for such kinds of system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of vector quasi-equilibrium problems and the generalized Debreu type equilibrium problem for vector-valued functions.     

The generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems

In this paper, the notion of the generalized Tykhonov well-posedness for system of vector quasi-equilibrium problems are investigated. By using the gap functions of the system of vector quasi-equilibrium problems, we establish the equivalent relationship between the generalized Tykhonov well-posedness of the system of vector quasi-equilibrium problems and that of the minimization problems. We also present some metric characterizations for the generalized Tykhonov well-posedness of the system of vector quasi-equilibrium problems. The results in this paper are new and extend some known results in the literature.     

Merit functions for general variational inequalities

In this paper, we consider some classes of merit functions for general variational inequalities. Using these functions, we obtain error bounds for the solution of general variational inequalities under some mild conditions. Since the general variational inequalities include variational inequalities, quasivariational inequalities and complementarity problems as special cases, results proved in this paper hold for these problems. In this respect, results obtained in this paper represent a refinement of previously known results for classical variational inequalities.     

Scalarization and Kuhn-Tucker-Like Conditions for Weak Vector Generalized Quasivariational Inequalities

We consider a weak vector generalized quasivariational inequality. By introducing a method of scalarization which does not require any assumption on the data and by using previous results of the authors concerning scalar generalized quasivariational inequalities, we present Kuhn-Tucker-like conditions for this problem in the case in which the set-valued operator of the constraints is defined by a finite number of inequalities     

Hybrid inclusion and disclusion systems with applications to equilibria and parametric optimization

In this paper, we first establish the existence theorems of the solution of hybrid inclusion and disclusion systems, from which we study mixed types of systems of generalized quasivariational inclusion and disclusion problems and systems of generalized vector quasiequilibrium problems. Some applications of existence theorems to feasible points for various mathematical programs with variational constraints or equilibrium constraints, system of vector saddle point and system of minimax theorem are also given.     

Existence of solutions for a generalized vector quasivariational inequality

The paper deals with a generalization of a vector quasivariational inequality. An existence theorem for its solutions is established; it is based on a kind of minimax inequality, which is here established for continuous affine mappings and differs from previous results. Fan's section theorem for set-valued mappings is extended. An application for an equilibrium problem of a network with vector-valued cost functions is given.This research was supported by the National Nature Science Foundation of China.     

带有模糊集值映象的一般混合拟变分包含的扰动近似点算法

本文引入一类新的带有模糊集值映象的一般混合拟变分包.在Hilbert空间中,利用极大η-单调映象的预解算子技巧,建立了这类变分包与不动点的等价性.利用这种等价性,构造了一些新的扰动近似点算法,并证明了由此算法所产生的迭代序列的收敛性.这些定理改进,统一和推广了近期文献中许多重要结果.