From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability

We consider a generalized equation governed by a strongly monotone and Lipschitz single-valued mapping and a maximally monotone set-valued mapping in a Hilbert space. We are interested in the sensitivity of solutions w.r.t. perturbations of both mappings. We demonstrate that the directional differentiability of the solution map can be verified by using the directional differentiability of the single-valued operator and of the resolvent of the set-valued mapping. The result is applied to quasi-generalized equations in which we have an additional dependence of the solution within the set-valued part of the equation.

Study on Generalized Directional Differentiability Problems of Fuzzy Mappings

This paper discusses the gH-directional differentiability of fuzzy mappings, and proposes the concept of gH-directional differentiability of fuzzy mappings. Based on the concept of gH-directional differentiability of interval-valued mappings and its related properties, two properties of gH-directional differentiability fuzzy mappings are proposed. At the same time, the relation between gH-differentiability and gH-directional differentiability for a fuzzy mapping is discussed, and it is proved that both gH-derivative and gH-partial derivative are directional derivatives of fuzzy mappings in the direction of the coordinate axis.

Hadamard Directional Differentiability of the Optimal Value Function of a Quadratic Programming Problem

In this paper, we consider the stability analysis of a convex quadratic programming (QP) problem and its restricted Wolfe dual when all parameters in the problem are perturbed. Based on the continuity of the feasible set mapping, we establish the upper semi-continuity of the optimal solution mappings of the convex QP problem and the restricted Wolfe dual problem. Furthermore, by characterizing the optimal value function as a min–max optimization problem over two compact convex sets, we demonstrate the Lipschitz continuity and the Hadamard directional differentiability of the optimal value function.