A Symplectic Discretization Based Proximal Point Algorithm for Convex Minimization

TL;DR

This paper introduces a novel symplectic discretization-based proximal point algorithm for convex minimization, providing convergence analysis via ODE systems and demonstrating its relation to existing accelerated methods.

Contribution

It develops the Symplectic Proximal Point Algorithm (SPPA) using symplectic Euler discretization and proves its convergence rate, linking it to existing accelerated algorithms.

Findings

The SPPA has a proven convergence rate.

Existing accelerated proximal point algorithms are special cases of SPPA.

Under additional assumptions, SPPA achieves a finer convergence rate.

Abstract

The proximal point algorithm plays a central role in non-smooth convex programming. The Augmented Lagrangian Method, one of the most famous optimization algorithms, has been found to be closely related to the proximal point algorithm. Due to its importance, accelerated variants of the proximal point algorithm have received considerable attention. In this paper, we first study an Ordinary Differential Equation (ODE) system, which provides valuable insights into proving the convergence rate of the desired algorithm. Using the Lyapunov function technique, we establish the convergence rate of the ODE system. Next, we apply the Symplectic Euler Method to discretize the ODE system to derive a new proximal point algorithm, called the Symplectic Proximal Point Algorithm (SPPA). By utilizing the proof techniques developed for the ODE system, we demonstrate the convergence rate of the SPPA. Additionally, it is shown that existing accelerated proximal point algorithm can be considered a special case of the SPPA in a specific manner. Furthermore, under several additional assumptions, we prove that the SPPA exhibits a finer convergence rate.