  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 $n$.
