dHPR: A Distributed Halpern Peaceman--Rachford Method for Non-smooth Distributed Optimization Problems

This paper introduces the distributed Halpern Peaceman–Rachford (dHPR) method, an efficient algorithm for solving distributed convex composite optimization problems with non-smooth objectives, which achieves a non-ergodic $O(1\mathrm{/}k)$ iteration complexity regarding Karush–Kuhn–Tucker residual. By leveraging the symmetric Gauss–Seidel decomposition, the dHPR effectively decouples the linear operators in the objective functions and consensus constraints while maintaining parallelizability and avoiding additional large proximal terms, leading to a decentralized implementation with provably fast convergence. The superior performance of dHPR is demonstrated through comprehensive numerical experiments on distributed LASSO, group LASSO, and $L_1$-regularized logistic regression problems.