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

Optimization of systems governed by variational inequalities with linear constraints in IR" 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 Clarke, 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. 1. FORMULATION OF THE PROBLEM AND CLASSICAL METHODS This paper contains a new approach to such optimization problems where the state is' governed by a variational inequality on a finite-dimensional space. For simplicity we confine ourselves to the elliptic case with control-independent linear constraints. Yet, our method could be extended to evolution variational inequalities (after a discretization both in space and in time) or to linear constraints depending on a control parameter as well. On the other hand, general convex constraints or nonlinear monotone operators in the variational inequality v/ould cause probably considerable complications. First we formulate our problem. Let bi e W, ct e U, i eIK, I K be a finite index set, denote the usual scalar product in IR", n ^ 1. We consider the convex poly­