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

Abstract

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.