Accelerated Decentralized Constraint-Coupled Optimization: A Dual${}^{2}$ Approach

Abstract

In this paper, we focus on a class of decentralized constraint-coupled optimization problem: $\mathbf{\min}_{\boldsymbol x_{\boldsymbol i}\in\mathbb{R}^{\boldsymbol d_{\boldsymbol i}},\boldsymbol{i\in\mathcal{I};y\in\mathbb{R}^{\boldsymbol p}}}$ $\sum_{\boldsymbol i = 1}^{\boldsymbol n}\left(\boldsymbol f_{\boldsymbol i}(\boldsymbol x_{\boldsymbol i})+\boldsymbol g_{\boldsymbol i}(\boldsymbol x_{\boldsymbol i})\right)+\boldsymbol h(\boldsymbol y)\ \mathbf{s.t.}\ \sum_{\boldsymbol i = 1}^{\boldsymbol n}\boldsymbol A_{\boldsymbol i}\boldsymbol x_{\boldsymbol i}=\boldsymbol{y}$, over an undirected and connected network of $\boldsymbol{n}$ agents. Here, $\boldsymbol{f_{i}}$, $\boldsymbol{g_{i}}$, and $\boldsymbol{A_{i}}$ represent private information of agent $\boldsymbol{i}\in\mathcal{I}=\{\mathbf{1,\cdots,}\boldsymbol{n}\}$, while $\boldsymbol{h}$ is public for all agents. Building on a novel dual${}^{\mathbf 2}$ approach, we develop two accelerated algorithms to solve this problem: the inexact Dual${}^{\mathbf 2}$ Accelerated (iD2A) gradient method and the Multi-consensus inexact Dual${}^{\mathbf 2}$ Accelerated (MiD2A) gradient method. We demonstrate that both iD2A and MiD2A can guarantee asymptotic convergence under a milder condition on $\boldsymbol{h}$ compared to existing algorithms. Furthermore, under additional assumptions, we establish linear convergence rates and derive significantly lower communication and computational complexity bounds than those of existing algorithms. Several numerical experiments validate our theoretical analysis and demonstrate the practical superiority of the proposed algorithms.