Optimality of Approximate Quasi-Weakly Efficient Solutions for Vector Equilibrium Problems via Convexificators

This paper is devoted to the investigation of optimality conditions for approximate quasi-weakly efficient solutions to a class of nonsmooth Vector Equilibrium Problem (VEP) via convexificators. First, a necessary optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is presented by making use of the properties of convexificators. Second, the notion of approximate pseudoconvex function in the form of convexificators is introduced, and its existence is verified by a concrete example. Under the introduced generalized convexity assumption, a sufficient optimality condition for approximate quasi-weakly efficient solutions to problem (VEP) is also established. Finally, a scalar characterization for approximate quasi-weakly efficient solutions to problem (VEP) is obtained by taking advantage of Tammer’s function.

Arcwise Connectedness of the Solution Sets for Generalized Vector Equilibrium Problems

In this research, by means of the scalarization method, arcwise connectedness results were established for the sets of globally efficient solutions, weakly efficient solutions, Henig efficient solutions and superefficient solutions for the generalized vector equilibrium problem under suitable assumptions of natural quasi cone-convexity and natural quasi cone-concavity.

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.

Error bound analysis for vector equilibrium problems with partial order provided by a polyhedral cone

The aim of this paper is to establish new results on the error bounds for a class of vector equilibrium problems with partial order provided by a polyhedral cone generated by some matrix. We first propose some regularized gap functions of this problem using the concept of $$\mathcal {G}_{A}$$ G A -convexity of a vector-valued function. Then, we derive error bounds for vector equilibrium problems with partial order given by a polyhedral cone in terms of regularized gap functions under some suitable conditions. Finally, a real-world application to a vector network equilibrium problem is given to illustrate the derived theoretical results.