A distributed augmented Lagrangian decomposition algorithm for constrained optimization

TL;DR

This work addresses large-scale constrained optimization with distributed data and coupling constraints. It introduces Distributed Augmented Lagrangian Decomposition ($DALD$), a solver-agnostic AL-based framework with inner and outer loops and a dedicated solver layer, underpinned by convergence proofs for the standard version. The paper extends DALD with accelerated variants and a hierarchical coordination network to unify and extend existing AL-based distributed methods, complemented by a thorough convergence analysis and practical experiments. Numerical results, including a non-convergence case for multi-block ADMM and a network flow problem, demonstrate robustness and efficiency gains from topology-aware coordination and inner-loop acceleration.

Abstract

Within the framework of the augmented Lagrangian (AL), we propose a novel distributed optimization method, termed Distributed Augmented Lagrangian Decomposition (DALD), and provide a rigorous convergence proof for its standard version. To address the high iteration costs in early stages, we propose several accelerated variants of DALD that enhances efficiency without compromising theoretical guarantees, supported by a comprehensive convergence analysis. To facilitate the description of the distributed optimization process, the concept of hierarchical coordination networks is introduced, integrating hierarchical matrix concepts to aid in this explanation. We further explore and expand the applicability of the DALD method and demonstrate how it unifies existing distributed optimization theories within the AL framework. The effectiveness and applicability of the proposed distributed optimization method and its variants are further validated through numerical experiments.

Figures (7)

• Figure 1: Distribution of decision-making variables and coupling relationships
• Figure 2: Hierarchical networks for describing the subproblem solving sequences
• Figure 3: Hierarchical matrices for describing the subproblem solving sequences
• Figure 4: A unified framework for distributed optimization
• Figure 5: Convergence trends for different values of $v_{\max}$