  In the article the author is studying the twice codifferentiable functions,
defined by Prof. V.Ph. Demyanov, and some methods for calculating their
codifferentials. At the beginning easier case is considered when a function is
twice hypodifferentiable. There is proved that a twice hypodifferentiable
positively homogeneous function $ h (\cdot) $ of the second order is maximum of
the quadratic forms with respect to a certain set of matrices, which coincides
with the convex hull of the limit matrices calculated at points, where the
original function $ h (\cdot) $ is twice differentiable, and these points tend
themselves to zero. It is shown that a set of the limit matrices coincides with
the second-order subdifferential, introduced by the author, of a positively
homogeneous function of the second order at the point zero. The author's first
and second subdifferentials are used to calculate the second codifferential of
a codifferentiable function $f(\cdot) $. The second hypodifferential and
hyperdifferential of a function $f(\cdot) $ are evaluated up to equivalence.
The proved theorems, that give the rules for calculating subdifferentials and
codifferentials, are important for practical optimization.
