Scalarization and decomposition of vector variational inequalities governed by bifunctions

Abstract

In this article we study the structure of solution sets within a special class of generalized Stampacchia-type vector variational inequalities, defined by means of a bifunction which takes values in a partially ordered Euclidean space. It is shown that, similar to multicriteria optimization problems, under appropriate convexity assumptions, the (weak) solutions of these vector variational inequalities can be recovered by solving a family of weighted scalar variational inequalities. Consequently, it is deduced that the set of weak solutions can be decomposed into the union of the sets of strong solutions of all variational inequalities obtained from the original one by selecting certain components of the bifunction which governs it.

Keywords:

• generalized vector variational inequality
• scalarization
• decomposition
• multicriteria optimization
• Pareto reducibility

AMS Subject Classifications:

• 49J40
• 90C29

Acknowledgements

N. Popoviei's research was partially supported by CNCS Project ID-2261 under Contract 543/2008.