Pseudomonotone $${_{\ast}}$$ maps and the cutting plane property

Abstract

Pseudomonotone $${_{\ast}}$$ maps are a generalization of paramonotone maps which is very closely related to the cutting plane property in variational inequality problems (VIP). In this paper, we first generalize the so-called minimum principle sufficiency and the maximum principle sufficiency for VIP with multivalued maps. Then we show that pseudomonotonicity $${_{\ast}}$$ of the map implies the "maximum principle sufficiency" and, in fact, is equivalent to it in a sense. We then present two applications of pseudomonotone $${_{\ast}}$$ maps. First we show that pseudomonotone $${_{\ast}}$$ maps can be used instead of the much more restricted class of pseudomonotone+ maps in a cutting plane method. Finally, an application to a proximal point method is given.