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

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