The role of strong convexity-concavity in the convergence and robustness of the saddle-point dynamics
This paper studies the projected saddle-point dynamics for a twice differentiable convex-concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We provide a novel characterization of the omega-limit set of the trajectories of these dynamics in terms of the diagonal Hessian blocks of the saddle function. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexity-concavity of the saddle function. If this property is global, and for the case when the saddle function takes the form of the Lagrangian of an equality constrained optimization problem, we establish the input-to-state stability of the saddle-point dynamics by providing an ISS Lyapunov function. Various examples illustrate our results.
© 2016 IEEE. We would like to thank Simon K. Niederländer for discussions on Lyapunov functions for the saddle-point dynamics. This work was supported by NSF award ECCS-1307176 and ARPA-e Cooperative Agreement DE-AR0000695 (AC and JC), NSF CPS grant CNS 1544771 (EM), and NSF CNS grant 1545096 (SL).