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

Xicai Deng,Wei Zhao

doi:10.1186/s13660-021-02666-z

Abstract

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.

Highlights

Stability analysis and sensitivity analysis have important theory and application in optimization theory
Li and Li [25] obtained some results on sensitivity analysis via a set-valued gap function of parametric vector equilibrium problems
We introduce a class of set-valued gap functions for parametric vector equilibrium problems

Summary

Introduction

Stability analysis and sensitivity analysis have important theory and application in optimization theory. The generalized derivatives and coderivatives for set-valued mappings have been used to study the sensitivity analysis of vector optimization problems. Li and Li [25] obtained some results on sensitivity analysis via a set-valued gap function of parametric vector equilibrium problems. The S-derivative and the set-valued gap function are exploited to study the sensitivity analysis of vector equilibrium problems. 3, we establish formulae of the S-derivative for a set-valued gap function of the parametric vector equilibrium problems. Proposition 2.1 The set-valued mapping V is a gap function of parametric vector equilibrium problems. We give an example on the class set-valued gap functions for parametric vector equilibrium problems. We derive the formulas for computing S-derivative of the set-valued gap function V for parametric vector equilibrium problems. Remark 3.1 We mention that our results in Theorem 3.1 and Theorem 3.2 are new, and they do not coincide with the existing ones in the literature (see [28,29,30] and the cited references therein)

S-derivative of perturbation maps for parametric vector equilibrium problems

Conclusion

Methods