The subdifferentials of the first and second orders for Lipschitz functions

TL;DR

This paper develops a unified theory of first and second order subdifferentials for Lipschitz functions using Steklov integrals, providing new rules, equivalences, and calculus for optimization applications.

Contribution

It introduces novel constructions of subdifferentials via Steklov integrals and proves their equivalence with existing definitions, extending the calculus of subdifferentials for optimization.

Findings

Subdifferentials of first order coincide with average integral limits of gradients.

Subdifferentials of second order match second derivatives for twice differentiable functions.

The paper provides necessary and sufficient optimality conditions using generalized gradients.

Abstract

Construction of an united theory of the subdifferentials of the first and second orders is interesting for many specialists in optimization \cite{morduchrockafel}. In the paper the rules for construction of the subdifferentials of the first and second orders are introduced. The constructions are done with the help of the Steklov integral of a Lipschitz function $f(\cdot)$ over the images of a set-valued mapping $D(\cdot)$. It is proved that the subdifferential of the first order consisting of the average integral limit values of the gradients $\nabla f(r(\cdot))$, calculated along the curves $r(\cdot)$ from an introduced set of curves $\eta$, coincides with the subdifferentials of the first order constructed using the Steklov integral introduced by the author for the first time in \cite{lowapp2}, \cite{lowapp2a}. If the function $f(\cdot)$ is twice differentiable at $x$ then the subdifferentials of the first and second orders coincide with the gradient $\nabla f(r(\cdot))$ and the matrix of the second mixed derivatives of $f(\cdot)$ at $x$. The generalized gradients and matrices are used for formulation of the necessary and sufficient conditions of optimality. The calculus for the subdifferentials of the first and second orders is constructed. The examples are given.