[Paper Review] Accelerated Hierarchical ADMM for Nonconvex Optimization in Multi-Agent Decision Making.

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

Distributed Control Multi-Agent Systems4 citations

TL;DR

This paper proposes a hierarchical three-block ADMM with accelerated barrier methods to solve nonconvex distributed optimization in multi-agent systems, particularly for UAV decision-making. By introducing slack variables and combining outer-loop ALM with inner-loop semi-proximal ADMM, the method achieves convergence to a stationary point and significantly improves computational efficiency over standard approaches.

ABSTRACT

Distributed optimization is widely used to solve large-scale problems effectively in a localized and coordinated manner. Thus, it is noteworthy that the methodology of distributed model predictive control (DMPC) has become a promising approach to achieve effective outcomes; and particularly in decision-making tasks for multi-agent systems. However, the typical deployment of such DMPC frameworks would lead to involvement of nonlinear processes with a large number of nonconvex constraints. Noting all these attendant constraints and limitations, the development and innovation of a hierarchical three-block alternating direction method of multipliers (ADMM) approach is presented in the work here to solve the nonconvex optimization problem that arises for such a decision-making problem in multi-agent systems. Firstly thus, an additional slack variable is introduced to relax the original large-scale nonconvex optimization problem; such that the intractable nonconvex coupling constraints are suitably related to the distributed agents. Then, the approach with a hierarchical ADMM that 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 shown that the appropriate desired stationary point exists for the procedures of the hierarchical stages for convergence in the algorithm. Next, the approximate optimization with a barrier method is then applied to accelerate the computational efficiency. Finally, a multi-agent system involving decision-making for multiple unmanned aerial vehicles (UAVs) is utilized to demonstrate the effectiveness of the proposed method in terms of attained performance and computational efficiency.

Motivation & Objective

• Address the challenge of solving large-scale nonconvex optimization problems in multi-agent decision-making with coupled constraints.
• Overcome the computational intractability of nonconvex coupling constraints in distributed model predictive control (DMPC) frameworks.
• Develop a scalable and convergent algorithm for nonconvex optimization that supports distributed agent coordination.
• Improve computational efficiency through barrier method acceleration in a hierarchical ADMM framework.
• Demonstrate the method’s effectiveness in a real-world multi-UAV decision-making scenario.

Proposed method

• Introduce a slack variable to relax the original nonconvex optimization problem, transforming intractable coupling constraints into distributed, manageable components.
• Implement a hierarchical ADMM structure with an outer loop using the augmented Lagrangian method (ALM) for global convergence.
• Employ an inner loop based on three-block semi-proximal ADMM to solve the relaxed subproblems in a distributed manner.
• Apply an approximate barrier method to accelerate convergence and enhance computational efficiency during optimization.
• Ensure convergence to a stationary point by leveraging the algorithmic structure and proper step-size selection.
• Decompose the problem across agents such that each solves local subproblems while coordinating via dual variables and consensus constraints.

Experimental results

Research questions

• RQ1 Can a hierarchical ADMM framework with three-block updates achieve convergence for nonconvex distributed optimization in multi-agent systems?
• RQ2 How does introducing a slack variable improve the tractability of nonconvex coupling constraints in distributed decision-making?
• RQ3 What is the impact of barrier method approximation on computational speed and solution quality in the proposed ADMM framework?
• RQ4 To what extent does the algorithm maintain convergence guarantees despite nonconvexity and distributed coordination?
• RQ5 How does the method perform in real-world multi-agent scenarios, such as UAV formation control?

Key findings

• The proposed hierarchical ADMM with ALM and semi-proximal updates converges to a stationary point under the given algorithmic structure.
• The introduction of slack variables successfully decouples nonconvex coupling constraints, enabling distributed solution procedures.
• The barrier method approximation significantly accelerates convergence, improving computational efficiency without sacrificing solution quality.
• The method demonstrates effective performance in a multi-UAV decision-making scenario, achieving both high solution quality and fast computation.
• The algorithm maintains stability and convergence even under nonconvex and large-scale problem conditions common in multi-agent systems.