We review the concept of VU-decomposition of nonsmooth convex functions, which is closely related to the notion of partly smooth functions. As VU-decomposition depends on the subdifferential at the given point, the associated objects lack suitable continuity properties (because the subdifferential lacks them), which poses an additional challenge to the already difficult task of constructing sup...

The Gauss-Lucas Theorem on the roots of polynomials nicely simplifies the computation of the subderivative and regular subdifferential of the abscissa mapping on polynomials (the maximum of the real parts of the roots). This paper extends this approach to more general functions of the roots. By combining the Gauss-Lucas methodology with an analysis of the splitting behavior of the roots, we obt...

It is known that the subdifferential of a lower semicontinuous convex function f over a Banach space X determines this function up to an additive constant in the sense that another function of the same type g whose subdifferential coincides with that of f at every point is equal to f plus a constant, i.e., g = f + c for some real constant c. Recently, Thibault and Zagrodny introduced a large cl...

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 ...

A brief story about late Professor Michel Bounias, a member of Advisory Board of Spacetime & Substance, is stated in the present note. His remarkable studies aimed at the understanding the constitution of the real space allow us to consider his findings as an actual breakthrough in the fundamental science. Bounias’ studies shed light on the organization of space arranged as a mathematical latti...

We study differential-geometrical structure of an information manifold equipped with a divergence function.A divergence function generates a Riemannian metric and furthermore it provides a symmetric third-order tensor,when the divergence is asymmetric. This induces a pair of affine connections dually coupled to each other withrespect to the Riemannian metric. This is the arising eme...

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...