Faster Stochastic ADMM for Nonsmooth Composite Convex Optimization in Hilbert Space

TL;DR

This paper introduces a stochastic ADMM algorithm tailored for nonsmooth convex optimization in Hilbert spaces, demonstrating faster convergence and practical efficiency for PDE-constrained problems.

Contribution

It proposes a novel stochastic ADMM method with proven strong convergence and improved nonergodic convergence rates for nonsmooth convex problems in Hilbert spaces.

Findings

Proven strong convergence in the strongly convex case

Faster nonergodic convergence rates for convex problems

Numerical results show the method's efficiency

Abstract

In this paper, a stochastic alternating direction method of multipliers (ADMM) is proposed for a class of nonsmooth composite and stochastic convex optimization problems in Hilbert space, motivated by optimization problems constrained by partial differential equation (PDE) with random coefficients. We prove the strong convergence of the proposed ADMM algorithm in the strongly convex case, and show the faster nonergodic convergence rates in terms of functional values and feasibility violation for both strongly convex and general convex cases. We demonstrate the application of the proposed method to solve certain model problems, along with its associated probability bound of large deviation. Some preliminary numerical results illustrate the efficiency of our method.