On the Stability of the Solution Sets of General Multivalued Vector Quasiequilibrium Problems

We give sufficient conditions for the semicontinuity of solution sets of general multivalued vector quasiequilibrium problems. All kinds of semicontinuities are considered: lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness. Moreover, we investigate the weak, middle, and strong solutions of quasiequilibrium problems. Many examples are provided to give more insights and comparisons with recent existing results.     

Optimality conditions for a unified vector optimization problem with not necessarily preordering relations

This paper studies a general vector optimization problem which encompasses those related to efficiency, weak efficiency, strict efficiency, proper efficiency and approximate efficiency among others involving non necessarily preordering relations. Based on existing results about complete characterization by scalarization of the solution set obtained by the same authors, several properties of (generalized) convexity and lower semicontinuity of the composition of the scalarizing functional and the objective vector function are studied. Finally, some optimality conditions are presented through subdifferentials in the convex and nonconvex case.     

一个标量泛函的研究及其在集值优化问题中的应用

本文在K条件下,研究了所给标量泛函的连续性和拟凸性,并利用该标量泛函,将集值优化问题转化为均衡问题,进而研究了含约束的集值优化问题弱充分解的存在性和拟集值优化问题强逼近解映射的上半连续性与下半连续性.与最近的文献相比,我们的方法是新的,条件和结论也更具一般性.     

参数弱向量平衡问题解集映射的连续性

运用非线性标量化方法, 讨论参数弱向量平衡问题解集映射的上半连续性和下半连续性, 并举例说明了所得结果的正确性.     

Semicontinuity and convergence for vector optimization problems with approximate equilibrium constraints

This paper mainly intends to present some semicontinuity and convergence results for perturbed vector optimization problems with approximate equilibrium constraints. We establish the lower semicontinuity of the efficient solution mapping for the vector optimization problem with perturbations of both the objective function and the constraint set. The constraint set is the set of approximate weak efficient solutions of the vector equilibrium problem. Moreover, upper Painlevé–Kuratowski convergence results of the weak efficient solution mapping are showed. Finally, some applications to the optimization problems with approximate vector variational inequality constraints and the traffic network equilibrium problems are also given. Our main results are different from the ones in the literature.     

Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems

In this paper we establish sufficient conditions for the solution set of parametric multivalued vector quasiequilibrium problems to be semicontinuous. All kinds of semicontinuity are considered: lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity and closedness. Moreover, we investigate both the “weak” and “strong” solutions of quasiequilibrium problems.     

Semicontinuity of Approximate Solution Mappings to Parametric Set-Valued Weak Vector Equilibrium Problems

In this article, we obtain some stability results for parametric weak vector equilibrium problem with set-valued mappings. By using a scalarization method, we establish sufficient conditions for the semicontinuity of the approximate solution mappings to parametric set-valued weak vector equilibrium problem under weak assumptions. These results extend and improve some known results in the literature.     

Lower Semicontinuity and Upper Semicontinuity of the Solution Sets and Approximate Solution Sets of Parametric Multivalued Quasivariational Inequalities

We consider the semicontinuity of the solution set and the approximate solution set of parametric multivalued quasivariational inequalities in topological vector spaces. Three kinds of problems arising from the multivalued situation are investigated. A rather complete picture, which is symmetric for the two kinds of semicontinuity (lower and upper semicontinuity) and for the three kinds of multivalued quasivariational inequality problems, is supplied. Moreover, we use a simple technique to prove the results. The results obtained improve several known ones in the literature. This research was partially supported by the National Basic Research Program in Natural Sciences of Vietnam. The final part of this work was completed during a stay of the first author at the Department of Mathematics, University of Pau, Pau, France, and its hospitality is acknowledged.     

Various kinds of semicontinuity and the solution sets of parametric multivalued symmetric vector quasiequilibrium problems

We introduce some definitions related to semicontinuity of multivalued mappings and discuss various kinds of semicontinuity-related properties. Sufficient conditions for the solution sets of parametric multivalued symmetric vector quasiequilibrium problems to have these properties are established. Comparisons of the solution sets of our two problems are also provided. As an example of applications of our main results, the mentioned semicontinuity-related properties of the solution sets to a lower and upper bounded quasiequilibrium problem are obtained as consequences.     

Semicontinuity of Solution Mappings to Parametric Generalized Vector Equilibrium Problems

In this article, stability results concerning the lower semicontinuity and the Hausdorff upper semicontinuity of the solution mappings to parametric generalized vector equilibrium problems with neither the monotonicity of mappings nor any information of the solution mappings are established by using scalarization methods and a new density result.     

Existence of Solutions of Systems of Generalized Implicit Vector Variational Inequalities

We consider five different types of systems of generalized vector variational inequalities and derive relationships among them. We introduce the concept of pseudomonotonicity for a family of multivalued maps and prove the existence of weak solutions of these problems under these pseudomonotonicity assumptions in the setting of Hausdorff topological vector spaces as well as real Banach spaces. We also establish the existence of a strong solution of our problems under lower semicontinuity for a family of multivalued maps involved in the formulation of the problems. By using a nonlinear scalar function, we introduce gap functions for our problems by which we can solve systems of generalized vector variational inequalities using optimization techniques. The first two authors were supported by SABIC and Fast Track Research Grants SAB-2006-05. They are grateful to the Department of Mathematical Sciences, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia for providing excellent research facilities.     

Sensitivity Analysis for Vector Equilibrium Problems under Functional Perturbations

This article is devoted to sensitivity analysis for vector equilibrium problems under functional perturbations. We show that the solution mapping is upper semicontinuous. Sufficient conditions for lower semicontinuity and Hölder continuity of the solution mapping are established. Finally, we derive some corollaries for special cases of vector equilibrium problems as examples.     

Semicontinuity of the Approximate Solution Sets of Multivalued Quasiequilibrium Problems

We consider two kinds of approximate solutions and approximate solution sets to multivalued quasiequilibrium problems. Sufficient conditions for the lower semicontinuity, Hausdorff lower semicontinuity, upper semicontinuity, Hausdorff upper semicontinuity, and closedness of these approximate solution sets are established. Applications in approximate quasivariational inequalities, approximate fixed points, and approximate quasioptimization problems are provided.     

Gap functions and error bounds for weak vector variational inequalities

In this article, by using the image space analysis, a gap function for weak vector variational inequalities is obtained. Its lower semicontinuity is also discussed. Then, these results are applied to obtain the error bounds for weak vector variational inequalities. These bounds provide effective estimated distances between a feasible point and the solution set of the weak vector variational inequalities.     

Tangency conditions for multivalued mappings

We prove that interiority conditions imply tangency conditions for two multivalued mappings from a topological space into a normed vector space. As a consequence, we obtain the lower semicontinuity of the intersection of two multivalued mappings. An application to the epi-upper semicontinuity of the sum of convex vector-valued mappings is given.     

Continuity and Convexity of a Nonlinear Scalarizing Function in Set Optimization Problems with Applications

Hern\(\acute{\mathrm{a}}\)ndez and Rodríguez-Marín (J Math Anal Appl 325:1–18, 2007) introduced a nonlinear scalarizing function for sets, which is a generalization of the Gerstewitz’s function. This paper aims at investigating some properties concerned with the nonlinear scalarizing function for sets. The continuity and convexity of the nonlinear scalarizing function for sets are showed under some suitable conditions. As applications, the upper semicontinuity and the lower semicontinuity of strongly approximate solution mappings to the parametric set optimization problems are also given.     

Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem

This paper deals with the stability for a parametric generalized strong vector equilibrium problem. Under new assumptions, which do not contain any information about the solution set and monotonicity, we establish the lower semicontinuity and upper semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem by using a scalarization method and a density result. These results are improve the corresponding ones in recent literature. Some examples are given to illustrate our results.     

New Scalarizing Approach to the Stability Analysis in Parametric Generalized Ky Fan Inequality Problems

This paper gives sufficient conditions for the upper and lower semicontinuities of the solution mapping of a parametric mixed generalized Ky Fan inequality problem. We use a new scalarizing approach quite different from traditional linear scalarization approaches which, in the framework of the stability analysis of solution mappings of equilibrium problems, were useful only for weak vector equilibrium problems and only under some convexity and strict monotonicity assumptions. The main tools of our approach are provided by two generalized versions of the nonlinear scalarization function of Gerstewitz. Our stability results are new and are obtained by a unified technique. An example is given to show that our results can be applied, while some corresponding earlier results cannot.     

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.     

含参弱向量均衡问题的适定性

在实Hausdorff拓扑线性空间中研究了含参弱向量均衡问题的适定性.证明了在适当条件下由近似网定义的含参适定性等价于近似解映射的上半连续性,并给出了所研究问题各种适定的充分性条件.