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Mathematical programs in which the constraint set is partially defined by the solutions of an elliptic variational inequality, so-called “elliptic MPECs,” are formulated in reflexive Banach spaces. With the goal of deriving explicit first-order optimality conditions amenable to the development of numerical procedures, variational analytic concepts are both applied and further developed. The paper is split into two main parts. The first part concerns the derivation of conditions in which the (lower-level) state constraints are assumed to be polyhedric sets. This part is then completed by two examples, the latter of which involves pointwise bilateral bounds on the gradient of the state. The second part focuses on an important nonpolyhedric example, namely, when the lower-level state constraints are presented by pointwise bounds on the Euclidean norm of the gradient of the state. A formula for the second-order (Mosco) epiderivative of the indicator function for this convex set is derived. This result is then used to demonstrate the (Hadamard) directional differentiability of the solution mapping of the variational inequality, which then leads to the derivation of explicit strong stationarity conditions for this problem.

First-Order Optimality Conditions for Elliptic Mathematical Programs with Equilibrium Constraints via Variational Analysis