Kurdyka-Łojasiewicz inequality of D-gap function for nonsmooth and nonmonotone variational-hemivariational inequality problem

In this paper, we investigate the difference gap function (for short, D-gap function) associated with a nonsmooth and nonmonotone variational-hemivariational inequality problem (for short, (VHVIP)). We derive explicit formulas for the subderivative, regular subdifferential, and limiting subdifferential of the D-gap function for such nonsmooth and nonmonotone (VHVIP). Utilizing these formulas, we establish the Kurdyka-Łojasiewicz inequality (for short, KŁ inequality) for the D-gap function of nonsmooth and nonmonotone (VHVIPs). Finally, we propose a descent algorithm for solving the (VHVIP) based on the D-gap function. As an application of our KŁ inequality result, we further prove the convergence of this algorithm.