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Based on concepts like kth convex hull and finer characterization ofnonconvexity of a function, we propose a refinement of the Shapley-Folkmanlemma and derive a new estimate for the duality gap of nonconvex optimizationproblems with separable objective functions. We apply our result to a networkflow problem and the dynamic spectrum management problem in communicationsystems as examples to demonstrate that the new bound can be qualitativelytighter than the existing ones. The idea is also applicable to cases withgeneral nonconvex constraints.

Refined Shapley-Folkman Lemma and Its Application in Duality Gap Estimation