Gap functions for vector variational inequalities

Abstract

In this article, we intend to study several scalar-valued gap functions for Stampacchia and Minty-type vector variational inequalities. We first introduce gap functions based on a scalarization technique and then develop a gap function without any scalarizing parameter. We then develop its regularized version and under mild conditions develop an error bound for vector variational inequalities with strongly monotone data. Further, we introduce the notion of a partial gap function which satisfies all, but one of the properties of the usual gap function. However, the partial gap function is convex and we provide upper and lower estimates of its directional derivative.

Keywords:

• vector variational inequalities
• gap functions
• error bounds

Acknowledgments

We are very grateful to the anonymous referees whose constructive suggestions have improved the presentation of the paper. The research of the author C. Charitha was supported by Grant DFG-SFB 755-TPA4, Germany.