Minimum and Maximum Principle Sufficiency for a Nonsmooth Variational Inequality

In this paper, the minimum and maximum principle sufficiency properties for a nonsmooth variational inequality problem (NVIP) are studied. We discuss the relationship among the solution set of an NVIP and those defined by its dual problem and relevant gap functions. For a pseudomonotone NVIP, the weaker sharpness of its solution set has been shown to be sufficient for it to have minimum principle sufficiency property. As special cases, pseudomonotonicity $$ _{*} $$ and pseudomonotonicity $$ ^{+} $$ of the relevant bifunction have been characterized, from which the minimum and maximum principle sufficiency properties have also been characterized.