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In this paper, we study the D-gap function associated with a nonsmooth andnonmonotone variational inequality problem. We present some exact formulas forthe subderivative, the regular subdifferential set, and the limitingsubdifferential set of the D-gap function. By virtue of these formulas, weprovide some sufficient and necessary conditions for the Kurdyka-{\L}ojasiewiczinequality property and the error bound property for the D-gap functions. As anapplication of our Kurdyka-{\L}ojasiewicz inequality result and the abstractconvergence result in [Attouch, et al., Convergence of descent methods forsemi-algebraic and tame problems: proximal algorithms, forward-backwardsplitting, and regularized Gauss-Seidel methods, Math. Program.,137(2013)91-129], we show that the sequence generated by a derivative freedescent algorithm with an inexact line search converges linearly to somesolution of the variational inequality problem.

Kurdyka-{\L}ojasiewicz Inequality and Error Bounds of D-Gap Functions for Nonsmooth and Nonmonotone Variational Inequality Problems