[Paper Review] Refined Shapely-Folkman Lemma and Its Application in Duality Gap Estimation

Yingjie Bi, Ao Tang|arXiv (Cornell University)|Oct 18, 2016

Sparse and Compressive Sensing Techniques9 references3 citations

TL;DR

This paper refines the Shapley-Folkman lemma by introducing kth convex hulls and a finer characterization of nonconvexity, enabling a tighter estimate of the duality gap in separable nonconvex optimization. The method improves existing bounds and is validated on network flow and dynamic spectrum management problems, showing qualitative improvements over prior estimates.

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.

Motivation & Objective

• To address the limitation of existing duality gap estimates in nonconvex optimization, particularly for separable problems with nonconvex objectives.
• To develop a refined version of the Shapley-Folkman lemma that captures finer structural properties of nonconvex functions.
• To provide a tighter, more informative bound on the duality gap using kth convex hulls and nonconvexity characterization.
• To demonstrate the applicability and superiority of the new bound in practical problems like network flow and dynamic spectrum management.

Proposed method

• Introduces the concept of the kth convex hull to refine the classical Shapley-Folkman lemma, enabling a more nuanced analysis of nonconvex sets.
• Proposes a finer characterization of nonconvexity in functions based on their deviation from convexity at different levels of aggregation.
• Derives a new duality gap estimate by combining the kth convex hull framework with the structure of separable nonconvex optimization problems.
• Applies the refined bound to specific problems, including network flow and dynamic spectrum management, to evaluate its tightness.
• Extends the framework to handle general nonconvex constraints beyond separable objectives.
• Uses the refined estimate to quantify the gap between the primal and dual optimal values in nonconvex settings.

Experimental results

Research questions

• RQ1 How can the classical Shapley-Folkman lemma be refined to better capture the nonconvexity of separable objective functions in optimization?
• RQ2 Can a tighter duality gap estimate be derived using kth convex hulls and a more detailed characterization of nonconvexity?
• RQ3 How does the new bound compare to existing duality gap estimates in practical nonconvex optimization problems?
• RQ4 To what extent can the refined lemma be extended to problems with general nonconvex constraints?

Key findings

• The refined Shapley-Folkman lemma provides a tighter duality gap estimate than existing methods by incorporating kth convex hulls and a detailed nonconvexity characterization.
• The new bound is qualitatively tighter than previous estimates in the context of network flow problems, improving the accuracy of duality gap prediction.
• In dynamic spectrum management, the refined bound yields a more precise estimation of the duality gap, suggesting improved performance in resource allocation.
• The method is extendable to optimization problems with general nonconvex constraints, broadening its applicability beyond separable cases.