Higher-order optimality conditions with an arbitrary non-differentiable function

TL;DR

This paper develops new higher-order derivatives and subdifferentials for non-differentiable functions, providing comprehensive optimality conditions and characterizations of local minima without restrictions on the functions.

Contribution

It introduces a harmonized higher-order directional derivative and subdifferential, extending optimality conditions to arbitrary non-differentiable functions and improving existing characterizations.

Findings

Necessary and sufficient conditions for local minima of order n

Introduction of higher-order critical directions

Extension of optimality conditions to non-differentiable functions

Abstract

In this paper, we introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is harmonized with the classical higher-order Fr\'echet directional derivative in the sense that both derivatives of the same order coincide if the last one exists. We obtain necessary and sufficient conditions of order $n$ ($n$ is a positive integer) for a local minimum and isolated local minimum of order $n$ of the given function in terms of these derivatives and subdifferentials. We do not require any restrictions on the function in our results. A notion of a higher-order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order $n$. Higher-order invex functions are defined. They are the largest class such that our necessary conditions for local minima are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order $n$.