Stability and instability in saddle point dynamics Part II: The subgradient method

TL;DR

This paper analyzes the asymptotic behavior of subgradient dynamics in saddle point problems within convex domains, revealing that their long-term solutions are characterized by linear ODEs and providing convergence criteria.

Contribution

It extends the analysis of saddle point dynamics to subgradient methods in convex domains, showing their omega-limit sets are solutions to linear ODEs and characterizing their asymptotic behavior.

Findings

Omega-limit sets are solutions to linear ODEs.

Subgradient dynamics on affine subspaces are smooth and well-characterized.

Various convergence criteria are established.

Abstract

In part I we considered the problem of convergence to a saddle point of a concave-convex function via gradient dynamics and an exact characterization was given to their asymptotic behaviour. In part II we consider a general class of subgradient dynamics that provide a restriction in an arbitrary convex domain. We show that despite the nonlinear and non-smooth 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.