Lower Complexity Bounds of First-Order Methods for Affinely Constrained Composite Nonconvex Problems

• 成果类型: Article; Early Access
• 署名作者: Liu, Wei; Lin, Qihang; Xu, Yangyang
• 署名单位: Rensselaer Polytechnic Institute; University of Iowa
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2023.0377
• 发表日期: 2025
• 关键词: alternating direction method; optimization; algorithm

摘要

Many recent studies on first-order methods (FOMs) focus on composite nonconvex nonsmooth optimization with linear and/or nonlinear function constraints. Upper (or worst-case) complexity bounds have been established for these methods. However, little can be claimed about their optimality, as no lower bound is known except for a few special smooth nonconvex cases. In this paper, we make the first attempt to establish lower complexity bounds of FOMs for solving a class of composite nonconvex nonsmooth optimization with linear constraints. Assuming two different first-order oracles, we establish lower complexity bounds of FOMs to produce a (near) c-stationary point of a problem (and its reformulation) in the considered problem class for any given tolerance c > 0. Our lower bounds indicate that the existence of a nonsmooth convex regularizer can evidently increase the difficulty of an affinely constrained regularized problem over its nonregularized counterpart. In addition, we show that our lower bound of FOMs with the second oracle is tight, with a difference of up to a logarithmic factor from an upper complexity bound established in a longer arXiv version of this work.