Self-adaptive ADMM for semi-strongly convex problems

TL;DR

This paper introduces a self-adaptive ADMM algorithm that dynamically updates the penalty parameter for semi-strongly convex problems, achieving accelerated convergence rates and robustness, with applications to broader problem classes.

Contribution

The paper proposes a novel self-adaptive ADMM that guarantees convergence with adaptive penalty updates, extending applicability to non-semi-strongly convex problems via a partial proximal point method.

Findings

Achieves accelerated convergence rate of O(1/k^2)

Demonstrates linear convergence and convergence of iteration points

Shows high efficiency and robustness in numerical experiments

Abstract

In this paper, we develop a self-adaptive ADMM that updates the penalty parameter adaptively. When one part of the objective function is strongly convex i.e., the problem is semi-strongly convex, our algorithm can update the penalty parameter adaptively with guaranteed convergence. We establish various types of convergence results including accelerated convergence rate of O(1/k^2), linear convergence and convergence of iteration points. This enhances various previous results because we allow the penalty parameter to change adaptively. We also develop a partial proximal point method with the subproblem solved by our adaptive ADMM. This enables us to solve problems without semi-strongly convex property. Numerical experiments are conducted to demonstrate the high efficiency and robustness of our method.