Partition-optimization with Schur convex sum objective functions

We study optimization problems over partitions of the finite set N = {1,..., n}, where each element i in the partitioned set N is associated with a real number θ(i) and the objective associated with a partition ρ = (π(1),..., π(p)) has the form F(π) = f(θ(π)), where θ(π) = (Σ(i∈π 1) θ(i),..., Σ(i∈π p) θ(i)). When F is to be either maximized or minimized, we obtain conditions that allow for simple constructions of partitions that are uniformly optimal for all Schur convex functions f.