The max-$k$-sum of a set of real scalars is the maximum sum of a subset of size $k$, or alternatively the sum of the $k$ largest elements. We study two extensions: First, we show how to obtain smooth approximations to functions that are pointwise max-$k$-sums of smooth functions. Second, we discuss how the max-$k$-sum can be defined on vectors in a finite-dimensional real vector space ordered by a closed convex cone.
manuscript, School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, September 2016 To appear in Mathematical Programming. The final publication is available at link.springer.com at https://link.springer.com/article/10.1007/s10107-017-1201-0