Connection between subdifferentials and codifferentials. Constructing the continuous codifferentials. I

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.