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

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 $n$.