Error bounds of regularized gap functions for nonsmooth variational inequality problems (Q879970)

The authors study the following variational inequality problem: to find \(x\in P\) such that\break \(\langle F(x^*),x- x^*\rangle\geq 0\), where \(P\) is a nonempty closed convex set in an Euclidean space \(\mathbb{R}^n\), \(F\) a locally Lipschitz mapping from \(P\) to \(\mathbb{R}^n\). It is well-known by \textit{J. H. Wu}, \textit{M. Florian} and \textit{P. Marcotte} [Math. Program., Ser. A 61, No. 3, 281--300 (1993; Zbl 0813.90111)] that \(x^*\) solves this problem if and only if its regularized gap function \(f_\gamma(x^*)= 0\) and \(x^*\) solves the minimization problem: \(\min\{f_\gamma(x): x\in P\}\). They investigate the Clarke-Rockafellar directional derivative of \(f_\gamma\), and show that \(\sqrt{f_\gamma}\) has an error bound on \(P\). They propose an algorithm of Armijo type and give a convergent result for it.