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