Higher-Order Optimality Conditions in Unconstrained Optimization with an Arbitrary Nondifferentiable Function

Vsevolod I. Ivanov

Published 2015-11-30Version 1

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 consistent 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 $n$ in terms of these derivatives and subdifferentials. We do not require any restrictions on the function in our results.