Let Kn be the Lorentz/second-order cone in IR. For any function f from IR to IR, one can define a corresponding vector-valued function f soc (x) on IR by applying f to the spectral values of the spectral decomposition of x ∈ IR with respect to Kn. It was shown by J.-S. Chen, X. Chen and P. Tseng that this vector-valued function inherits from f the properties of continuity, Lipschitz continuity,...

With every real valued function, of a real argument, can be associated a matrix function mapping a linear space of symmetric matrices into itself. In this paper we study directional differentiability properties of such matrix functions associated with directionally differentiable real valued functions. In particular, we show that matrix valued functions inherit semismooth properties of the corr...

Abstract We focus on elliptic quasi-variational inequalities (QVIs) of obstacle type and prove a number results the existence solutions, directional differentiability optimal control such QVIs. give three theorems based an order approach, iteration scheme sequential regularisation through partial differential equations. show that solution map taking source term into set solutions QVI is directi...

In this paper we study the behavior of the local solutions of perturbed variational inequalities, governed by perturbations to both the variational inequality function and the feasible region. Assuming appropriate second-order and regularity conditions, we show that the perturbed local solution set is nonempty, Lipschitz continuous, and directionally differentiable. Even when the directional di...

Studying a critical value function φ in parametric nonlinear programming, we recall conditions guaranteeing that φ is a C function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of Dφ. Several specializations and applications are discussed. These results are understood as supplements to the well–developed theory of f...