[Paper Review] Hierarchical ADMM for Nonconvex Cooperative Distributed Model Predictive Control

Xiaoxue Zhang, Jun Ma+4 more|arXiv (Cornell University)|Nov 1, 2020

Distributed Control Multi-Agent Systems27 references4 citations

TL;DR

This paper proposes a hierarchical three-block ADMM approach to solve nonconvex cooperative distributed model predictive control (DMPC) in multi-agent systems. By introducing a slack variable to relax the nonconvex problem and combining outer-loop augmented Lagrangian method with inner-loop semi-proximal ADMM, the method achieves convergence to a stationary point and improves computational efficiency via a barrier method, demonstrating strong performance in UAV coordination tasks.

ABSTRACT

Distributed optimization is often widely attempted and innovated as an attractive and preferred methodology to solve large-scale problems effectively in a localized and coordinated manner. Thus along this line, it is noteworthy that the methodology of distributed model predictive control (DMPC) has become a promising approach to achieve effective outcomes, e.g., in decision-making tasks for multi-agent systems. However, the typical deployment of such distributed MPC frameworks would lead to the involvement of nonlinear processes with a large number of nonconvex constraints. To address this important problem, the development and innovation of a hierarchical three-block alternating direction method of multipliers (ADMM) approach is presented in this work to solve this nonconvex cooperative DMPC problem in multi-agent systems. Here firstly, an additional slack variable is introduced to relax the original large-scale nonconvex optimization problem. Then, a hierarchical ADMM approach, which contains outer loop iteration by the augmented Lagrangian method (ALM) and inner loop iteration by three-block semi-proximal ADMM, is utilized to solve the resulting relaxed nonconvex optimization problem. Additionally, it is analytically shown and established that the requisite desired stationary point exists for the procedures of the hierarchical stages for convergence in the algorithm. Finally, an approximate optimization stage with a barrier method is then applied to further significantly improve the computational efficiency, yielding the final improved hierarchical ADMM. The effectiveness of the proposed method in terms of attained performance and computational efficiency is demonstrated on a cooperative DMPC problem of decision-making process for multiple unmanned aerial vehicles (UAVs).

Motivation & Objective

• To address the challenge of solving large-scale nonconvex optimization problems in cooperative distributed model predictive control (DMPC) for multi-agent systems.
• To overcome the limitations of standard ADMM in handling nonconvex constraints and nonlinearity in DMPC frameworks.
• To develop a hierarchical algorithm that ensures convergence to a stationary point in nonconvex settings.
• To improve computational efficiency through a barrier method approximation in the optimization process.
• To validate the method’s effectiveness in real-world multi-UAV coordination scenarios.

Proposed method

• Introduces a slack variable to relax the original large-scale nonconvex optimization problem into a more tractable form.
• Employs a hierarchical ADMM framework with an outer loop using the augmented Lagrangian method (ALM) for global coordination.
• Uses an inner loop based on three-block semi-proximal ADMM to solve subproblems in a distributed and coordinated manner.
• Applies a barrier method in an approximate optimization stage to enhance computational efficiency without sacrificing convergence guarantees.
• Establishes analytical convergence to a stationary point by proving the existence of required optimality conditions in the hierarchical stages.
• Coordinates the iterative updates across agents using dual decomposition and penalty-based relaxation to maintain distributed structure.

Experimental results

Research questions

• RQ1 Can a hierarchical ADMM framework effectively handle nonconvex constraints in cooperative distributed model predictive control?
• RQ2 Does the proposed method guarantee convergence to a stationary point in nonconvex optimization settings?
• RQ3 How does the integration of a barrier method improve computational efficiency in the hierarchical ADMM framework?
• RQ4 What is the performance and scalability of the method in coordinating multiple unmanned aerial vehicles (UAVs)?
• RQ5 Can the three-block semi-proximal ADMM strategy maintain convergence and distributed computation in nonconvex DMPC problems?

Key findings

• The proposed hierarchical ADMM approach successfully converges to a stationary point in the nonconvex cooperative DMPC problem, as analytically established.
• The use of a slack variable enables effective relaxation of the original nonconvex problem, making it amenable to decomposition.
• The integration of the barrier method significantly enhances computational efficiency in the final optimization stage.
• The method demonstrates strong performance in coordinating multiple UAVs, achieving effective decision-making in cooperative tasks.
• The hierarchical structure maintains distributed computation while ensuring convergence, even under nonconvex and nonlinear constraints.
• The algorithm preserves the distributed nature of DMPC while improving robustness and scalability in multi-agent systems.