Constrained composite optimization and augmented Lagrangian methods

• 成果类型: Article; Early Access
• 署名作者: De Marchi, Alberto; Jia, Xiaoxi; Kanzow, Christian; Mehlitz, Patrick
• 署名单位: Bundeswehr University Munich; University of Wurzburg; Brandenburg University of Technology Cottbus
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-022-01922-4
• 发表日期: 2023
• 关键词: algorithm

摘要

We investigate finite-dimensional constrained structured optimization problems, featuring composite objective functions and set-membership constraints. Offering an expressive yet simple language, this problem class provides a modeling framework for a variety of applications. We study stationarity and regularity concepts, and propose a flexible augmented Lagrangian scheme. We provide a theoretical characterization of the algorithm and its asymptotic properties, deriving convergence results for fully nonconvex problems. It is demonstrated how the inner subproblems can be solved by off-the-shelf proximal methods, notwithstanding the possibility to adopt any solvers, insofar as they return approximate stationary points. Finally, we describe our matrix-free implementation of the proposed algorithm and test it numerically. Illustrative examples show the versatility of constrained composite programs as a modeling tool and expose difficulties arising in this vast problem class.