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

TL;DR

This paper introduces a refined version of the Shapley-Folkman lemma to improve duality gap estimates in nonconvex optimization, demonstrating tighter bounds in network flow and spectrum management problems.

Contribution

It presents a new refinement of the Shapley-Folkman lemma and derives sharper duality gap bounds for nonconvex problems with separable objectives.

Findings

Tighter duality gap bounds than existing estimates.

Application to network flow and spectrum management problems.

Applicable to general nonconvex constraints.

Abstract

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.