BALPA: A Balanced Primal-Dual Algorithm for Nonsmooth Optimization with Application to Distributed Optimization

TL;DR

This paper introduces BALPA, a primal-dual algorithm for nonsmooth composite optimization that balances primal and dual updates, improving efficiency especially when certain norms are large, with applications to distributed optimization.

Contribution

The paper presents BALPA, a novel primal-dual proximal splitting algorithm with a balanced dual update, and extends it to a stochastic version for distributed optimization.

Findings

BALPA achieves fast convergence for large norm problems.

S-BALPA effectively handles stochastic and distributed settings.

Numerical experiments confirm the efficiency of the proposed algorithms.

Abstract

In this paper, we propose a novel primal-dual proximal splitting algorithm (PD-PSA), named BALPA, for the composite optimization problem with equality constraints, where the loss function consists of a smooth term and a nonsmooth term composed with a linear mapping. In BALPA, the dual update is designed as a proximal point for a time-varying quadratic function, which balances the implementation of primal and dual update and retains the proximity-induced feature of classic PD-PSAs. In addition, by this balance, BALPA eliminates the inefficiency of classic PD-PSAs for composite optimization problems in which the Euclidean norm of the linear mapping or the equality constraint mapping is large. Therefore, BALPA not only inherits the advantages of simple structure and easy implementation of classic PD-PSAs but also ensures a fast convergence when these norms are large. Moreover, we propose a stochastic version of BALPA (S-BALPA) and apply the developed BALPA to distributed optimization to devise a new distributed optimization algorithm. Furthermore, a comprehensive convergence analysis for BALPA and S-BALPA is conducted, respectively. Finally, numerical experiments demonstrate the efficiency of the proposed algorithms.