Accelerated Convergence of Saddle-Point Dynamics
In this thesis, a second-order continuous-time saddle-point dynamics is introduced that mimics Nesterov's accelerated gradient flow dynamics. We study the convergence properties of this dynamics using a family of time-varying Lyapunov functions. In particular, we study the convergence rate of the dynamics for classes of strongly convex-strongly concave functions. For a class of quadratic strongly convex-strongly concave functions and under appropriate assumptions, this dynamics achieves global asymptotic convergence; in fact, further conditions lead to an accelerated convergence rate. We also provide conditions for both local asymptotic convergence and local accelerated convergence of general strongly convex-strongly concave functions.