Parametric Vector Equilibrium Problems Research Articles

Continuity of the solution set mappings to parametric unified weak vector equilibrium problems via free-disposal sets

In this paper, the lower and upper semicontinuity of the solution mappings for parametric unified weak vector equilibrium problem (PUWVEP) are established, via freedisposal set and nonlinear scalarization function. Moreover, example is given to illustrate the obtained results. The results improve the corresponding ones in the literature [12-14].

S-derivative of perturbed mapping and solution mapping for parametric vector equilibrium problems

In this paper, we deal with the sensitivity analysis in vector equilibrium problems by using the S-derivative of a set-valued mapping. We first investigate the S-derivative on a kind of set-valued gap function for the vector equilibrium problems. Based on these results, S-derivative estimations on a perturbed mapping for the parametric vector equilibrium problem are given. Moreover, we provide some examples to illustrate the obtained results. Finally, we derive the S-derivative estimations of a solutions mapping of the parametric vector equilibrium problems via S-derivative estimations of a kind of the parametric variational system.

Semicontinuity of the solution maps to vector equilibrium problems with equilibrium constraints

ABSTRACT In this paper, we are concerned with parametric vector equilibrium problems with equilibrium constraints. We first introduce concepts of generalized semicontinuity and convexity of vector-valued maps in ordered spaces and discuss their properties, as well as their relationships with well-known concepts. Next, employing mentioned concepts of generalized semicontinuity of vector-valued maps, we study sufficient conditions of upper semicontinuity for solution maps to such problems. Then, combining these generalized semicontinuity along with generalized convexity assumptions, sufficient conditions for lower semicontinuity of solution maps to the reference problems are established. Several examples are provided to illustrate the relaxation and essentialness of assumptions. Our results are new, even for special cases of the underlying problems.

Upper order-preservation of the solution correspondence to vector equilibrium problems

In this paper, several order-theoretic fixed point theorems are proved on partially ordered topological spaces. Applying these fixed point theorems, we explore the existence and upper order-preservation for parametric vector equilibrium problems. In contrast to the previous results on vector equilibrium problems, the upper order-preservation of solutions is a new subject, which would be useful for predicting the changing trend of solutions to vector equilibrium problems. In addition, neither topological continuity nor convexity of the considered vector-valued bifunction F is required in our results.

Stability of approximating solutions to parametric bilevel vector equilibrium problems and applications

In this paper, we establish sufficient conditions for the approximate solution mappings of parametric bilevel equilibrium problems with stability properties such as upper semicontinuity, lower semicontinuity, Hausdorff lower semicontinuity, continuity and Hausdorff continuity. Moreover, we also apply these results to parametric traffic network problems with equilibrium constraints. Many examples are provided to ensure the essentialness of the assumptions.

Strong vector equilibrium problems with LSC approximate solution mappings

This paper introduces two classes of parametric strong vector equilibrium problems whose approximate solution mappings are lower semicontinuous. In the first class, the objective set-valued maps satisfy some cone-convexity/cone-concavity assumptions, and in the second one, they satisfy some strongly proper cone-quasiconvexconcavity assumptions. All these mentioned concepts of generalized cone-convexity/cone-concavity/ strongly proper cone-quasiconvexconcavity are new and different from the traditional ones. Some upper semicontinuity/continuity results are also obtained. Applications to parametric weak u-set and l-set optimization problems and weak vector multivalued equilibrium problems are given.

Upper Semicontinuity of Parametric Lexicographic Vector Equilibrium Problems

In this paper, lexicographic vector equilibrium problems are investigated. By using the idea of sequential process, the upper semicontinuity and closedness of the solution set map are established for a parametric lexicographic strong vector equilibrium problem.

Stability of solution mappings for parametric bilevel vector equilibrium problems

In this paper, we first revisit the parametric bilevel vector equilibrium problems in Hausdorff topological vector spaces. Then we study the stability conditions such as (Hausdorff) upper semicontinuity, (Hausdorff) lower semicontinuity, outer-continuity and outer-openness of solutions for such problems. Many examples are provided to illustrate the essentialness of the imposed assumptions. For the applications, we obtain the stability results for the parametric vector variational inequality problems with equilibrium constraints and parametric vector optimization problems with equilibrium constraints.

Lower semicontinuity of solutions for order-perturbed parametric vector equilibrium problems

The lower semicontinuity of the (weak) efficient solution mappings for parametric vector equilibrium problems under more weaker assumptions is established. Some examples are developed to illustrate our results are real generalization different from recent ones in the literature and to describe the essential conditions of the latest results in the references are not real essential.

Lower Semicontinuity Results in Parametric Multivalued Weak Vector Equilibrium Problems and Applications

ABSTRACTThis article gives new sufficient conditions for the lower semicontinuity of the solution mapping of a parametric multivalued weak vector equilibrium problem with moving cones. A scalarizing approach, based on the signed distance function of Hiriart Urruty is used to discuss this lower semicontinuity property. The main results of the article are obtained under some assumptions different from those introduced earlier by previous linear and nonlinear scalarizing approaches. Some applications to the study of connectedness of weak solution sets of multivalued vector equilibrium problems are given.

Lipschitz continuity of an approximate solution mapping for parametric set-valued vector equilibrium problems

The purpose of this paper is to generalize and improve some topological properties of solutions set to the set-valued vector equilibrium problems by using the scalar characterization method. Moreover, the Lipschitz continuity of an approximate solution mapping for the parametric set-valued vector equilibrium problems is studied.

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.

A Note on Continuity of Solution Set for Parametric Weak Vector Equilibrium Problems

We consider the parametric weak vector equilibrium problem. By using a weaker assumption of Peng and Chang (2014), the sufficient conditions for continuity of the solution mappings to a parametric weak vector equilibrium problem are established. Examples are provided to illustrate the essentialness of imposed assumptions. As advantages of the results, we derive the continuity of solution mappings for vector optimization problems.

Stability of solution mapping for parametric symmetric vector equilibrium problems

This paper is concerned with the stability for a parametric symmetric vector equilibrium problem. A parametric gap function for the parametric symmetric vector equilibrium problem is introduced and investigated. By virtue of this function, we establish the sufficient and necessary conditions for the Hausdorff lower semicontinuity of solution mapping to a parametric symmetric vector equilibrium problem. The results presented in this paper generalize and improve the corresponding results in the recent literature.

On parametric implicit vector variational inequality problems

Recently Huang et al. (Math. Comput. Model. 43:1267-1274, 2006)introduced a class of parametric implicit vector equilibrium problems (for shortPIVEP) and they presented some existence results for a solution of PIVEP. Also,they provided two theorems about upper and lower semi-continuity of the solutionset of PIVEP in a locally convex Hausdorff topological vector space. The paperextends the corresponding results obtained in the setting of topological vectorspaces with mild assumptions and removing the notion of locally non-positivenessat a point and lower semi-continuity of the parametric mapping.

Hölder Continuity of Solutions to Parametric Vector Equilibrium Problems with Nonlinear Scalarization

In this article, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of solution mappings for both set-valued and single-valued cases to parametric vector equilibrium problems is studied. The nonlinear scalarization function is a powerful tool that plays a key role in the proofs, and its main properties (such as sublinearity, continuity, convexity) are fully employed. Especially, its locally and globally Lipschitz properties are provided and the Lipschitz property is first exploited to investigate the Hölder continuity of solutions.

Notes on Lipschitz Properties of Nonlinear Scalarization Functions with Applications

Various kinds of nonlinear scalarization functions play important roles in vector optimization. Among them, the one commonly known as the Gerstewitz function is good at scalarizing. In linear normed spaces, the globally Lipschitz property of such function is deduced via primal and dual spaces approaches, respectively. The equivalence of both expressions for globally Lipschitz constants obtained by primal and dual spaces approaches is established. In particular, when the ordering cone is polyhedral, the expression for calculating Lipschitz constant is given. As direct applications of the Lipschitz property, several sufficient conditions for Hölder continuity of both single-valued and set-valued solution mappings to parametric vector equilibrium problems are obtained using the nonlinear scalarization approach.

参数弱向量平衡问题解集的似Holder性和相依导数

In this paper, we study the Holder-likeness and contingent derivative of solutions to a parametric weak vector equilibrium problem. At first, recalling a real-valued gap function of the problem, we discuss the Lipschitz continuity and the Hadamard directional differentiability of the gap function. Then by means of the properties of the gap function we establish the Holder-likeness and Holder continuity and an explicit expression for the contingent derivative of the solution mapping for the problem, respectively.

A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems

By using a new assumption, the lower semicontinuity of the efficient solution mappings for parametric vector equilibrium problems without a monotonicity assumption is established. Our result is different from recent ones in the literature. Some examples are given to illustrate the case.

Semicontinuity of the solution mappings to weak generalized parametric Ky Fan inequality problems with trifunctions

In this article, we discuss the lower semicontinuity of solution maps without the condition of C-strict monotonicity for two classes of weak generalized parametric Ky Fan inequalities under the case that the f-solution set be a general set-valued one. Our results extend the recent ones in the literature (e.g. Cheng, Y.H., Zhu, D.L.: Global stability results for the weak vector variational inequality. J. Glob. Optim. 32, 543–550 (2005); C.R. Chen and S.J. Li, On the solution continuity of parametric generalized systems, Pac. J. Optim. 6 (2010), pp. 141–151; Gong, X.H., Yao, J.C.: Lower semicontinuity of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 197–205 (2008); Gong, X.H.: Continuity of the solution set to parametric weak vector equilibrium problems. J. Optim. Theory Appl. 139, 35–46 (2008)). Several examples are given for the illustration of our results.