Frank-Wolfe meets Shapley-Folkman: a systematic approach for solving nonconvex separable problems with linear constraints

TL;DR

This paper introduces a systematic two-stage approach for solving nonconvex separable optimization problems with linear constraints, leveraging duality bounds and primal solution construction via Caratheodory representations.

Contribution

It develops a novel method combining dual approximation and primal solution trimming, extending classical duality gap bounds to nonconvex domains with tractable Fenchel conjugates.

Findings

Recovers classical duality gap bounds for convex functions

Extends duality bounds to nonconvex functions

Provides a tractable approach for primal solution construction

Abstract

We consider separable nonconvex optimization problems under affine constraints. For these problems, the Shapley-Folkman theorem provides an upper bound on the duality gap as a function of the nonconvexity of the objective functions, but does not provide a systematic way to construct primal solutions satisfying that bound. In this work, we develop a two-stage approach to do so. The first stage approximates the optimal dual value with a large set of primal feasible solutions. In the second stage, this set is trimmed down to a primal solution by computing (approximate) Caratheodory representations. The main computational requirement of our method is tractability of the Fenchel conjugates of the component functions and their (sub)gradients. When the function domains are convex, the method recovers the classical duality gap bounds obtained via Shapley-Folkman. When the function domains are nonconvex, the method also recovers classical duality gap bounds from the literature, based on a more general notion of nonconvexity.