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

TL;DR

This work presents a constructive framework for solving nonconvex separable optimization with affine constraints by marrying Shapley–Folkman duality with Carathéodory representations. A two-stage method first approximates the optimal dual value via a large set of primal points using Frank–Wolfe with Fenchel-conjugate LDOs, then trims to a conic Carathéodory representation to produce primal feasible solutions that meet duality-gap bounds, even when domains are nonconvex. It provides exact and approximate Carathéodory reconstruction schemes, analyzes convex and general-domain cases, and extends to perturbation-based feasibility guarantees. The approach is demonstrated on Unit Commitment and PEV charging problems, showing practical primal feasibility and tight duality-gap behavior with scalable computation. Overall, the paper delivers a systematic, constructive pathway to leverage Shapley–Folkman bounds for realistic, large-scale nonconvex optimization tasks.

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.

Figures (3)

• Figure 1: Running time of the different algorithms with respect to $n$, averaged over 5 runs. (Left) Running time of \ref{['alg:FW1']}. (Middle) Running time of all three Carathéodory approaches. We see that the exact approach (\ref{['alg:Constructive-caratheodory']}) has long running time even for small values of $n$. (Right) Running time of the approximate Carathéodory approaches for larger values of $n$. We see that as $n$ grows, the MNP algorithm becomes more efficient than the FCFW algorithm.
• Figure 2: Infeasibility of the final reconstructed solution $\bar{x}_K$ with respect to the number of iterations $K$ of \ref{['alg:FW1']}, averaged over 10 runs. (Left) Infeasibility of the solution in the perturbed primal problem. In red is just the (scaled) curve $1/\sqrt{K}$ for reference. We see that the infeasibility seems to be decreasing close to the $O(1/\sqrt{K})$ rate as predicted by our bounds, and the slightly worse behavior is most likely due to the additive exponential error term $\text{Err}_A(n, T)$. (Right) Infeasibility of the solution in the original primal problem. We see that over 10 runs, it was always sufficient to run for only $K=1000$ iterations to obtain a fully feasible solution.
• Figure 3: Typical behavior of our method on two specific instances of the PEVs problem. The curve in blue represents the value of the function at the final reconstructed solution $\bar{x}_K$ with respect to the number of iterations $K$ of \ref{['alg:FW1']}. In orange is the value obtained by the randomized algorithm from udell2016bounding. The vertical dashed red line indicates the first value of $K$ for which $\bar{x}_K$ is primal feasible, i.e. $A \bar{x}_K - b \leq 0$.

Theorems & Definitions (36)

• Proposition 2.1
• Theorem 2.1: Carathéodory
• Theorem 2.2: Conic Carathéodory
• Theorem 2.3
• Lemma 2.1
• Theorem 2.4: Shapley Folkman
• Lemma 2.2
• Theorem 2.5
• Theorem 2.6
• remark thmcounterremark