Kurdyka-Łojasiewicz Inequality and Error Bounds of D-Gap Functions for Nonsmooth and Nonmonotone Variational Inequality Problems

In this paper, we study regularized/D-gap functions associated with a nonsmooth and nonmonotone variational inequality problem. We present some exact formulas for the subderivative, the regular subdifferential, and the limiting subdifferential of the regularized/D-gap functions respectively. By virtue of these formulas, we provide some sufficient conditions and necessary conditions for the Kurdyka-Łojasiewicz inequality property and the error bound property for the D-gap function respectively. As an application of our Kurdyka-Łojasiewicz inequality result, we show that, under certain mild assumptions, the sequence generated by a derivative-free descent algorithm with an inexact line search converges linearly to a solution of the variational inequality problem.