Semiglobal exponential stability of the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm for constrained optimization

• 成果类型: Article
• 署名作者: Bin, Michelangelo; Notarnicola, Ivano; Parisini, Thomas
• 署名单位: University of Bologna; Imperial College London; University of Cyprus; University of Trieste
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-02051-2
• 发表日期: 2024
• 页码: 629-660
• 关键词: Distributed optimization; DYNAMICS; CONVERGENCE

摘要

We consider the discrete-time Arrow-Hurwicz-Uzawa primal-dual algorithm, also known as the first-order Lagrangian method, for constrained optimization problems involving a smooth strongly convex cost and smooth convex constraints. We prove that, for every given compact set of initial conditions, there always exists a sufficiently small stepsize guaranteeing exponential stability of the optimal primal-dual solution of the problem with a domain of attraction including the initialization set. Inspired by the analysis of nonlinear oscillators, the stability proof is based on a non-quadratic Lyapunov function including a nonlinear cross term.