Gap functions and error bounds for variational-hemivariational inequalities

In this paper we investigate the gap functions and regularized gap functions for a class of variational–hemivariational inequalities of elliptic type. First, based on regularized gap functions introduced by Yamashita and Fukushima, we establish some regularized gap functions for the variational–hemivariational inequalities. Then, the global error bounds for such inequalities in terms of regularized gap functions are derived by using the properties of the Clarke generalized gradient. Finally, an application to a stationary nonsmooth semipermeability problem is given to illustrate our main results

Gap functions for quasi-equilibria

An approach for solving quasi-equilibrium problems (QEPs) is proposed relying on gap functions, which allow reformulating QEPs as global optimization problems. The (generalized) smoothness properties of a gap function are analysed and an upper estimate of its Clarke directional derivative is given. Monotonicity assumptions on both the equilibrium and constraining bifunctions are a key tool to guarantee that all the stationary points of a gap function actually solve QEP. A few classes of constraints satisfying such assumptions are identified covering a wide range of situations. Relying on these results, a descent method for solving QEP is devised and its convergence proved. Finally, error bounds are given in order to guarantee the boundedness of the sequence generated by the algorithm