Gap functions for constrained vector variational inequalities with applications

Abstract

In this paper, the image space analysis is applied to investigate scalar-valued gap functions and their applications for a (parametric)-constrained vector variational inequality. Firstly, using a non-linear regular weak separation function in image space, a gap function of a constrained vector variational inequality is obtained without any assumptions. Then, as an application of the gap function, two error bounds for the constrained vector variational inequality are derived by means of the gap function under some mild assumptions. Further, a parametric gap function of a parametric constrained vector variational inequality is presented. As an application of the parametric gap function, a sufficient condition for the continuity of the solution map of the parametric constrained vector variational inequality is established within the continuity and strict convexity of the parametric gap function. These assumptions do not include any information on the solution set of the parametric constrained vector variational inequality.

Keywords:

• Constrained vector variational inequality
• gap function
• error bound
• continuity
• image space analysis

Acknowledgements

The authors would like to express their sincere thanks to the anonymous referee and the Associate Editor Tim Hoheisel for helpful comments and suggestions which have vastly improved the presentation of the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [grant number 11426055], [grant number 11601054], the Basic and Advanced Research Project of CQ CSTC [grant number cstc2014jcyjA00044], [grant number cstc2016jcyjA00466], [grant number cstc2016jcyjA0163] and the Science and Technology Research Project of Chongqing Municipal Education Commission [grant number KJ1500419], [grant number KJ1600403], [grant number KJ1600433].