A general formula for the horizon function of a convex composite function

Abstract.

In many variational problems the behavior of functions or sets at infinity is of crucial importance. An appropriate concept for dealing with this topic has been introduced and widely studied under the name of recession analysis. The terminology is not unique: the recession cones (resp. functions) are also called asymptotic cones (resp. functions) or horizon cones (resp. functions). In the convex case calculus rules are available for the horizon function of a sum or a max of two functions, of a marginal function, of the composite of a convex function with an affine operator. The purpose of this paper is to provide a general formula for the composite of an extended real-valued convex function with a convex mapping with values in an ordered topological vector space. It seems that this problem has never been tackled in full generality. Some preliminaries about lower semicontinuous mappings and convex composite functions are necessary. Various examples are given.