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.

中文翻译：

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非光滑变分不等式的最小和最大原理充分性

本文研究了非光滑变分不等式问题（NVIP）的最小和最大原理充分性。我们讨论了NVIP解决方案集及其对偶问题和相关差距函数所定义的解决方案之间的关系。对于伪单调NVIP，已证明其解决方案集的较弱清晰度足以使其具有最小的原则自足性。作为特殊情况，已对相关双功能的伪单调性\（_ {*} \）和伪单调性\（^ {+} \）进行了特征化，从中也描述了最小和最大原理自足性。