The aim of the present paper is to provide a formula for the ε subdifferential of f +g ◦h different from the ones which can be found in the existent literature. Further we equivalently characterize this formula by using a so-called closedness type regularity condition expressed by means of the epigraphs of the conjugates of the functions involved. Even more, using the ε subdifferential formula ...

The main purpose of these lectures is to familiarize the student with the basic ingredients of convex analysis, especially its subdifferential calculus. This is done while moving to a clearly discernible end-goal, the Karush-Kuhn-Tucker theorem, which is one of the main results of nonlinear programming. Of course, in the present lectures we have to limit ourselves most of the time to the Karush...

We propose a level proximal subdifferential for proper lower semicontinuous function. Level is uniform refinement of the well-known subdifferential, and has pleasant feature that its resolvent always coincides with mapping It turns out representation in terms Mordukhovich limiting only valid hypoconvex functions. also provide properties numerous examples to illustrate our results.

We study properties of functions convex with respect to a given family X of vector fields, a notion that appears natural in Carnot-Carathéodory metric spaces. We define a suitable subdifferential and show that a continuous function is X -convex if and only if such subdifferential is nonempty at every point. For vector fields of Carnot type we deduce from this property that a generalized Fenchel...

In this note we give a subdifferential mean value inequality for every continuous Gâteaux subdifferentiable function f in a Banach space which only requires a bound for one but not necessarily all of the subgradients of f at every point of its domain. We also give a subdifferential approximate Rolle’s theorem stating that if a subdifferentiable function oscillates between −ε and ε on the bounda...

In this paper we study the optimal control of system driven by hemivariational inequality of second order. First, we establish the existence of solutions to hemivariational inequality which contains nonlinear pseudomonotone evolution operator. Introducing a control variable in the multivalued term of the generalized subdifferential, we prove the closedness (in suitable topologies) of the graph ...

Using a quantitative version of the subdifferential characterization of directionally Lipschitz functions, we study the integrability of subdifferentials of such functions over arbitrary Banach space.