On the Convex Hull of the Composition of a Separable and a Linear Function

Theorems on the convex hull of an extended real-valued function living on a Hilbert space are presented in case the function is separable and in case the function is a composition of another function and a linear transformation. Equivalence of convex hulls of functions and their biconjugates is used. In particular, it is shown that under properness conditions the biconjugate of a separable function is equal to the sum of the biconjugates of its constituents, and the biconjugate of a composition of a function and a bounded linear transformation is equal to the composition of the biconjugate of the function and the linear transformation on the range of the linear transformation. The results are applied to describe the convex hull of the objective function of a problem in stochastic integer programming.