  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.
