Convergence Results of Two-Step Inertial Proximal Point Algorithm

TL;DR

This paper introduces a two-step inertial proximal point algorithm with proven weak convergence and an $O(1/n)$ rate, demonstrating accelerated performance in convex optimization applications.

Contribution

It presents a novel inertial proximal point algorithm with convergence guarantees and applies it to well-known convex optimization methods.

Findings

Weak convergence of the algorithm is established.

The algorithm achieves an $O(1/n)$ non-asymptotic convergence rate.

Numerical results show accelerated performance over existing methods.

Abstract

This paper proposes a two-point inertial proximal point algorithm to find zero of maximal monotone operators in Hilbert spaces. We obtain weak convergence results and non-asymptotic $O(1/n)$ convergence rate of our proposed algorithm in non-ergodic sense. Applications of our results to various well-known convex optimization methods, such as the proximal method of multipliers and the alternating direction method of multipliers are given. Numerical results are given to demonstrate the accelerating behaviors of our method over other related methods in the literature.