Treffer: Evaluation of Clarke's generalized gradient in optimization of variational inequalities

Summary: Optimization of systems governed by variational inequalities with linear constraints in \({\mathbb{R}}^ n\) yields nonsmooth cost functions to be minimized. After discussion about various numerical methods to solve such optimization problems, we propose usage of a bundle or a subgradient algorithm and deal with the only problem how to evaluate in such case the generalized gradient of \textit{F. H. Clarke} [Optimization and nonsmooth analysis (1983; Zbl 0582.49001)] required by it. As the problem in general is very complicated, an effective procedure is proposed only for certain special data. However, using the transversality theory, it is shown that ``almost all'' (possibly in the generic sense) sufficiently smooth data fulfil the conditions that guarantee the validity of the procedure proposed.