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Pseudomonotone\(_*\) maps and the cutting plane property. (English) Zbl 1173.49010

Summary: Pseudomonotone\(_*\) 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\(_*\) 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\(_*\) maps. First we show that pseudomonotone\(_*\) 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.

MSC:

49J40 Variational inequalities
47H04 Set-valued operators
65K05 Numerical mathematical programming methods
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