Accelerated forward-backward and Douglas-Rachford splitting dynamics
This work addresses optimization efficiency for nonsmooth composite problems, offering incremental improvements in convergence analysis for specific algorithms.
The paper tackles the convergence of continuous-time accelerated Forward-Backward and Douglas-Rachford splitting algorithms for nonsmooth composite optimization, establishing accelerated sublinear and exponential convergence rates for convex and strongly convex problems, with computational experiments demonstrating effectiveness.
We examine convergence properties of continuous-time variants of accelerated Forward-Backward (FB) and Douglas-Rachford (DR) splitting algorithms for nonsmooth composite optimization problems. When the objective function is given by the sum of a quadratic and a nonsmooth term, we establish accelerated sublinear and exponential convergence rates for convex and strongly convex problems, respectively. Moreover, for FB splitting dynamics, we demonstrate that accelerated exponential convergence rate carries over to general strongly convex problems. In our Lyapunov-based analysis we exploit the variable-metric gradient interpretations of FB and DR splittings to obtain smooth Lyapunov functions that allow us to establish accelerated convergence rates. We provide computational experiments to demonstrate the merits and the effectiveness of our analysis.