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

TL;DR

This paper investigates the mathematical properties of the D-gap function related to nonsmooth, nonmonotone variational inequalities, establishing conditions for error bounds and convergence of descent algorithms.

Contribution

It provides new formulas for subdifferentials of the D-gap function and links the Kurdyka-{\

Findings

Derived explicit formulas for subderivatives and subdifferentials of the D-gap function.

Established necessary and sufficient conditions for Kurdyka-{\

showed linear convergence of a derivative-free descent algorithm for solving variational inequalities.

Abstract

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.