Directional Differentiability, Coexhausters, Codifferentials and Polyhedral DC Functions

Abstract

Codifferentials and coexhausters are used to describe nonhomogeneous approximations of a nonsmooth function. Despite the fact that coexhausters are modern generalizations of codifferentials, the theories of these two concepts continue to develop simultaneously. Moreover, codifferentials and coexhausters are strongly connected with DC functions. In this paper we trace analogies between all these objects, and prove the equivalence of the boundedness and optimality conditions described in terms of these notions. This allows one to extend the results derived in terms of one object to the problems stated via the other one. Another contribution of this paper is the study of connection between nonhomogeneous approximations and directional derivatives and formulate optimality conditions in terms of nonhomogeneous approximations. Introduction Among the variety of approaches of nonsmooth analysis [1] the method of quasidifferential stands out due to its constructiveness. One important advantage of this approach is that all the tools and methods can be built and used not only theoretically but also in practical problems. The approach goes back to the early 80-th when Demyanov, Rubinov and Polyakova proposed and studied the notion of quasidifferentials [2–5]. Quasidifferentials are pairs of convex compact sets that enable one to represent the directional derivative of a function at a point in a form of sum of maximum and minimum of a linear functions. Quasidifferentials enjoy full calculus, that grants the calculation of quasidifferentials for a rich variety of functions. Such functions are also called quasidifferentiable. Polyakova and Demyanov derived optimality conditions in terms of these objects and also showed how to find the directions of steepest descent and ascent when these conditions are not satisfied. This paved a way for constructing new