The hypodifferential and the $\varepsilon$-subdifferential of polyhedral function

The class of polyhedral functions is the simplest among the family of nonsmooth functions. One of the basic concepts of convex analysis is the notion of $\varepsilon$-subdifferential. The $\varepsilon$-subdifferential mapping is continuous in the Hausdorff metric. This property is used in the construction of continuous optimization methods for convex functions. The notions of hypodifferential and continuous hypodifferential were introduced by V. F. Demyanov. A polyhedron of special form can be taken as a continuous hypodifferential for a polyhedral function. In the paper, the properties of this hypodifferential and the $\varepsilon$-subdifferential of the polyhedral function are discussed. The relationship between them is established. A geometric interpretation of the hypodifferential is derived and the examples illustrating application of the developed theory are presented.