Calmness and Calculus: Two Basic Patterns

We establish two types of estimates for generalized derivatives of set-valued mappings which carry the essence of two basic patterns observed throughout the pile of calculus rules. These estimates also illustrate the role of the essential assumptions that accompany these two patters, namely calmness on the one hand and (fuzzy) inner calmness* on the other. Afterwards, we study the relationship between and sufficient conditions for the various notions of (inner) calmness. The aforementioned estimates are applied in order to recover several prominent calculus rules for tangents and normals as well as generalized derivatives of marginal functions and compositions as well as Cartesian products of set-valued mappings under mild conditions. We believe that our enhanced approach puts the overall generalized calculus into some other light. Some applications of our findings are presented which exemplary address necessary optimality conditions for minimax optimization problems as well as the calculus related to the recently introduced semismoothness* property.

Error bounds for overdetermined and underdetermined generalized centred simplex gradients

Using the Moore–Penrose pseudoinverse this work generalizes the gradient approximation technique called the centred simplex gradient to allow sample sets containing any number of points. This approximation technique is called the generalized centred simplex gradient. We develop error bounds and, under a full-rank condition, show that the error bounds have ${\mathcal O}(\varDelta ^2)$, where $\varDelta $ is the radius of the sample set of points used. We establish calculus rules for generalized centred simplex gradients, introduce a calculus-based generalized centred simplex gradient and confirm that error bounds for this new approach are also ${\mathcal O}(\varDelta ^2)$. We provide several examples to illustrate the results and some benefits of these new methods.

Calculus Rules forV-Proximal Subdifferentials in Smooth Banach Spaces

In 2010, Bounkhel et al. introduced new proximal concepts (analytic proximal subdifferential, geometric proximal subdifferential, and proximal normal cone) in reflexive smooth Banach spaces. They proved, inp-uniformly convex andq-uniformly smooth Banach spaces, the density theorem for the new concepts of proximal subdifferential and various important properties for both proximal subdifferential concepts and the proximal normal cone concept. In this paper, we establish calculus rules (fuzzy sum rule and chain rule) for both proximal subdifferentials and we prove the Bishop-Phelps theorem for the proximal normal cone. The limiting concept for both proximal subdifferentials and for the proximal normal cone is defined and studied. We prove that both limiting constructions coincide with the Mordukhovich constructions under some assumptions on the space. Applications to nonconvex minimisation problems and nonconvex variational inequalities are established.