Consensus Planning with Primal, Dual, and Proximal Agents

TL;DR

The paper addresses coordinating decision making across heterogeneous, distributed agents by introducing a generic Consensus Planning Protocol (CPP) that permits primal, dual, and proximal interfaces. It proposes the 3-Agent Consensus Planning (3ACP) algorithm, which blends ADMM-like updates for proximal agents, dual ascent for dual agents, and linearized ADMM for primal agents, and proves sublinear convergence $O\left(\frac{1}{k}\right)$ with two-step linear convergence under stronger assumptions. The authors provide a detailed convergence analysis, including plain convergence, sublinear ergodic rates, and linear rates, as well as practical enhancements such as acceleration and tighter quadratic bounds. They also demonstrate mixed-quadratic agent scenarios to illustrate update structures and empirical behavior, highlighting robustness to agent mix and the potential for scalable coordination in complex supply chains. Overall, the work enables flexible, principled coordination in large systems without requiring modification of existing agent interfaces, with clear pathways for improvement and extension to asynchronous settings.

Abstract

Consensus planning is a method for coordinating decision making across complex systems and organizations, including complex supply chain optimization pipelines. It arises when large interdependent distributed agents (systems) share common resources and must act in order to achieve a joint goal. In prior consensus planning work, all agents have been assumed to have the same interaction pattern (e.g., all dual agents or all primal agents or all proximal agents), most commonly using the Alternating Direction Method of Multipliers (ADMM) as proximal agents. However, this is often not a valid assumption in practice, where agents consist of large complex systems, and where we might not have the luxury of modifying these large complex systems at will. In this paper, we introduce a consensus algorithm that overcomes this hurdle by allowing for the coordination of agents with different types of interfaces (named primal, dual, and proximal). Our consensus planning algorithm allows for any mix of agents by combining ADMM-like updates for the proximal agents, dual ascent updates for the dual agents, and linearized ADMM updates for the primal agents. We prove convergence results for the algorithm, namely a sublinear O(1/k) convergence rate under mild assumptions, and two-step linear convergence under stronger assumptions. We also discuss enhancements to the basic method and provide illustrative empirical results.

Figures (1)

• Figure 1: Relative error of the consensus plan iterates for different configurations of the mix of agents and learning rates: $\rho_p=\rho_d=\rho_x=0.1$ (top left), $\rho_p=\rho_d=\rho_x=1$ (top right), $\rho_p=\rho_x=10$, $\rho_d=1$ (bottom left), $\rho_p=\rho_x=50$, $\rho_d=1$ (bottom right).

Theorems & Definitions (22)

• Remark 3.1
• Proposition 4.4
• proof
• Proposition 4.5
• proof
• Remark 4.6
• Theorem 4.7: Convergence of Algorithm \ref{['alg:firstalgo']}
• proof
• Theorem 4.8: Ergodic Sublinear Convergence
• proof