Analytical and Computational Advances in Quasidifferential Calculus for Nonsmooth Optimization
Alberto Peretti, C. Sutti · 2024
In this paper, after giving some basic definitions about quasidifferentiability ( Sect.1 ), in Sect.2 we prove a necessary and sufficient condition for the sub differentiability of a one-dimensional real function, by making also some geometrical considerations. In Sect.3 we introduce the concept of directional subdifferentiability; a relationship between subdifferentiability and directional subdifferentiability is found, together with a way to express the directional sub differential in terms of subdifferential. In Sect.4 we study what connections exist between convexity, convexity of level sets, Hessian matrix and sub differentiability, by means of example functions. It is shown in detail that the convexity of the directional derivative is necessary and sufficient for the subdifferentiability of a directionally differentiable function. Each result is illustrated with many significant examples.