On a generalization of paramonotone maps and its application to solving the Stampacchia variational inequality

Abstract

Paramonotone maps are monotone maps that satisfy a mild additional condition. They were introduced in order to ensure convergence of certain algorithms for solving the Stampacchia variational inequality. Recently, this concept has been generalized to single-valued pseudomonotone_* maps. In this article, we extend this definition to multivalued pseudomonotone_* maps. We show that this new class of maps includes the subdifferentials of locally Lipschitz pseudoconvex functions. In addition, it is shown to be exactly the class of pseudomonotone maps that have a certain cutting plane property, thus ensuring convergence of the above-mentioned algorithms. We give a specific example of a perturbed auxiliary problem method that leads to a solution of the Stampacchia variational inequality with a multivalued, pseudomonotone_* map.

Keywords:

• Paramonotone maps
• Pseudomonotone maps
• Variational inequality problems

Keywords:

• Mathematics Subject Classifications 2000: 49J40
• 49J53
• 47H05