Stability and instability in saddle point dynamics Part II: The subgradient method
Provides theoretical convergence guarantees for subgradient methods in saddle point problems, relevant for optimization and game theory.
This paper analyzes subgradient dynamics for saddle point problems in convex domains, showing that their ω-limit set consists of solutions to linear ODEs, enabling convergence criteria based on prior results.
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 $ω$-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.