On concepts of directional differentiability
Journal of Optimization Theory and Applications
An implicit-function theorem for a class of nonsmooth functions
Mathematics of Operations Research
SIAM Journal on Control and Optimization
Sensitivity of Solutions to Variational Inequalities on Banach Spaces
SIAM Journal on Control and Optimization
Convex analysis and variational problems
Convex analysis and variational problems
Mathematical Programs with Complementarity Constraints: Stationarity, Optimality, and Sensitivity
Mathematics of Operations Research
Optimal Control of Bilateral Obstacle Problems
SIAM Journal on Control and Optimization
Computational Methods for Option Pricing (Frontiers in Applied Mathematics) (Frontiers in Applied Mathematics 30)
Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization (Mps-Siam Series on Optimization 6)
A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs
Computational Optimization and Applications
Computational Optimization and Applications
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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.