BALPA: A Balanced Primal-Dual Algorithm for Nonsmooth Optimization with Application to Distributed Optimization
This work addresses inefficiencies in distributed optimization for applications with large constraint norms, offering an incremental improvement over existing primal-dual methods.
The paper tackles the problem of composite optimization with equality constraints by proposing BALPA, a primal-dual algorithm that balances primal and dual updates to improve convergence speed, especially when linear mapping norms are large, and demonstrates its efficiency through numerical experiments.
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.