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

• 成果类型: Article; Early Access
• 署名作者: Dubois-Taine, Benjamin; d'Aspremont, Alexandre
• 署名单位: Universite PSL; Ecole Normale Superieure (ENS); Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-025-02271-8
• 发表日期: 2025
• 关键词: unit commitment; point

摘要

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) Carath & eacute;odory 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.