  Based on concepts like kth convex hull and finer characterization of
nonconvexity of a function, we propose a refinement of the Shapley-Folkman
lemma and derive a new estimate for the duality gap of nonconvex optimization
problems with separable objective functions. We apply our result to a network
flow problem and the dynamic spectrum management problem in communication
systems as examples to demonstrate that the new bound can be qualitatively
tighter than the existing ones. The idea is also applicable to cases with
general nonconvex constraints.
