Asymptotically tight Lagrangian dual of smooth nonconvex problems via improved error bound of Shapley-Folkman Lemma
In convex geometry, the Shapley-Folkman Lemma asserts that the nonconvexity of a Minkowski sum of $n$ dimensional bounded nonconvex sets does not accumulate once the number of summands exceeds the dimension $n$, and thus the sum becomes approximately convex. Originally published by Starr in the context of quasi-equilibrium in nonconvex market models in economics, the lemma has since found widespread use in optimization, particularly for estimating the duality gap of the Lagrangian dual of separable nonconvex problems. Given its foundational nature, we pose the following geometric question: Is it possible for the nonconvexity of the Minkowski sum of $n$-dimensional nonconvex sets to even diminish instead of just not accumulating as the number of summands increases, under some general conditions? We answer this affirmatively. First, we provide an elementary geometric proof of the Shapley-Folkman Lemma based on the facial structure of the convex hull of each set. This leads to refinement of the classical error bound derived from the lemma. Building on this new geometric perspective, we further show that when most of the sets satisfy a certain local smoothness condition which naturally arises in the epigraphs of smooth functions, their Minkowski sum converges directly to a convex set, with a vanishing nonconvexity measure. In optimization, this implies that the Lagrangian dual of block-structured smooth nonconvex problems with potentially additional sparsity constraints is asymptotically tight under mild assumptions, which contracts nonvanishing duality gap obtained via classical Shapley-Folkman Lemma.