Haihang Lan | Search arXiv

arXiv2026

A Distributed Accelerated Preconditioned ADMM for Solving Quadratically Regularized Optimal Transport on Bipartite Graphs

Optimal transport (OT) has found broad applications in machine learning, economics, and management. Classical OT models, however, are not directly suited to allocation problems with graph-structured connectivity and unbalanced mass. Distributed OT formulations on bipartite graphs address these requirements but are typically solved by ADMM, whose slow convergence limits communication efficiency. We adapt the accelerated preconditioned ADMM (AP-ADMM) to the quadratically regularized distributed OT model and develop DAP-ADMM, a fully distributed algorithm in which each node communicates only with its graph neighbors. For fixed local penalty parameters, we establish global convergence and non-ergodic $O(1/k)$ bounds for both the KKT-residual norm and the primal objective gap. We derive an exact solver-free routine for the interval-constrained projections arising in the local ADMM subproblems. We propose a distributed self-adaptive penalty mechanism that updates node-specific penalties using locally computable KKT-residual components without global aggregation. Numerical experiments confirm the efficiency of the solver-free routine and the practical acceleration of DAP-ADMM over standard distributed ADMM.

cs.DC↗

arXiv2026

An Inexact Halpern-accelerated Preconditioned Generalized Proximal Point Algorithm for the Maximal Monotone Inclusion Problem

This paper studies an inexact Halpern-accelerated generalized proximal point algorithm with an admissible positive semidefinite preconditioner $\mathcal{M}$ for solving maximal monotone inclusion problems. The proposed framework combines Halpern anchoring, resolvent inexactness, and relaxation over the full weight range $ρ\in(0,2]$, thereby covering both under-relaxed and over-relaxed proximal iterations within a unified scheme. We first establish convergence of the inexact resolvent sequence under conditions that allow general anchoring parameters and certain nonsummable tolerances. For anchoring parameters $β_{k}=1/(k+r)$ with $r\geq2$, we then derive explicit bounds on the squared fixed-point residual in the $\mathcal{M}$-seminorm. In particular, if the tolerances satisfy $\varepsilon_{k}=\mathcal{O}((k+1)^{-α})$ with $α>3/2$ for $0<ρ<2$ and $α>2$ for $ρ=2$, these bounds yield an $\mathcal{O}(1/k^{2})$ convergence rate. The stronger decay condition on the inexactness tolerances at $ρ=2$ highlights a qualitative distinction between the endpoint and the interior regime in the inexact setting. Finally, we develop inexact accelerated versions of the preconditioned alternating direction method of multipliers (pADMM) and the preconditioned primal--dual hybrid gradient (PDHG) method based on this framework, derive inexactness criteria based on subproblem residuals, and establish a nonergodic $\mathcal{O}(1/k)$ KKT residual rate for their inexact iterates.

math.OC↗