Connection between subdifferentials and codifferentials. Constructing the continuous codifferentials. I

TL;DR

This paper explores the relationship between subdifferentials and codifferentials for twice codifferentiable functions, providing methods for calculating second-order codifferentials and their applications in optimization.

Contribution

It introduces new methods for calculating second-order codifferentials using subdifferentials, extending the theoretical framework for optimization of nonsmooth functions.

Findings

Proved that twice hypodifferentiable positively homogeneous functions are maxima of quadratic forms.

Identified the second-order subdifferential as the set of limit matrices.

Developed practical rules for calculating subdifferentials and codifferentials.

Abstract

In the article the author is studying the twice codifferentiable functions, defined by Prof. V.Ph. Demyanov, and some methods for calculating their codifferentials. At the beginning easier case is considered when a function is twice hypodifferentiable. There is proved that a twice hypodifferentiable positively homogeneous function $ h (\cdot) $ of the second order is maximum of the quadratic forms with respect to a certain set of matrices, which coincides with the convex hull of the limit matrices calculated at points, where the original function $ h (\cdot) $ is twice differentiable, and these points tend themselves to zero. It is shown that a set of the limit matrices coincides with the second-order subdifferential, introduced by the author, of a positively homogeneous function of the second order at the point zero. The author's first and second subdifferentials are used to calculate the second codifferential of a codifferentiable function $f(\cdot) $. The second hypodifferential and hyperdifferential of a function $f(\cdot) $ are evaluated up to equivalence. The proved theorems, that give the rules for calculating subdifferentials and codifferentials, are important for practical optimization.