Stability and Instability in Saddle Point Dynamics Part II: The Subgradient Method

• 成果类型: Article
• 署名作者: Holding, Thomas; Lestas, Ioannis
• 署名单位: Imperial College London; University of Cambridge
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2020.3019381
• 发表日期: 2021
• 页码: 2945-2960
• 关键词: Nonlinear systems; nonsmooth systems; networks; optimization; saddle points; subgradient dynamics

摘要

In part I we considered the problem of convergence to a saddle point of a concave-convex function in C-2 via gradient dynamics and an exact characterization was given to their asymptotic behavior. In part II we consider a general class of subgradient dynamics that provide a restriction in a convex domain. We show that despite the nonlinear and nonsmooth character of these dynamics their omega-limit set is comprised of solutions to only linear ODEs. In particular, we show that the latter are solutions to subgradient dynamics on affine subspaces which is a smooth class of dynamics the asymptotic properties of which have been exactly characterized in part I. Various convergence criteria are formulated using these results and several examples and applications are also discussed throughout the manuscript.