Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems (2408.15969v2)

Abstract: We examine stability properties of primal-dual gradient flow dynamics for composite convex optimization problems with multiple, possibly nonsmooth, terms in the objective function under the generalized consensus constraint. The proposed dynamics are based on the proximal augmented Lagrangian and they provide a viable alternative to ADMM which faces significant challenges from both analysis and implementation viewpoints in large-scale multi-block scenarios. In contrast to customized algorithms with individualized convergence guarantees, we develop a systematic approach for solving a broad class of challenging composite optimization problems. We leverage various structural properties to establish global (exponential) convergence guarantees for the proposed dynamics. Our assumptions are much weaker than those required to prove (exponential) stability of primal-dual dynamics as well as (linear) convergence of discrete-time methods such as standard two-block and multi-block ADMM and EXTRA algorithms. Finally, we show necessity of some of our structural assumptions for exponential stability and provide computational experiments to demonstrate the convenience of the proposed approach for parallel and distributed computing applications.

• The paper introduces novel primal-dual gradient flow dynamics leveraging the proximal augmented Lagrangian framework for multi-block convex optimization.
• The analysis demonstrates global exponential convergence of the proposed dynamics under weaker assumptions compared to existing methods like ADMM.
• Numerical experiments showcase the practicality and effectiveness of these dynamics for large-scale, distributed computing problems across various applications.

Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems

The paper "Stability of Primal-Dual Gradient Flow Dynamics for Multi-Block Convex Optimization Problems" investigates the convergence properties of primal-dual gradient flow dynamics in the context of composite convex optimization. This paper introduces novel primal-dual dynamics based on the proximal augmented Lagrangian framework for solving composite convex optimization problems consisting of multiple nonsmooth terms subjected to a generalized consensus constraint. Unlike traditional approaches that often face significant analytical and practical challenges, especially in large-scale, multi-block settings, this research provides a systematic and robust alternative.

The focus is on guaranteeing global stability and exponential convergence of the proposed dynamics. Analysis shows that the stability and convergence properties of these dynamics can be established under weaker assumptions than those traditionally required by methods like the Alternating Direction Method of Multipliers (ADMM). The ADMM, while popular, has limitations in multi-block scenarios that this approach overcomes.

Key Contributions and Findings

1. Proximal Augmented Lagrangian: The authors introduce primal-dual gradient flow dynamics leveraging the proximal augmented Lagrangian formulation, enabling the decomposition of complex optimization tasks into simpler sub-problems. This decomposition facilitates parallel and distributed computing, a significant advantage in handling large-scale problems.
2. Convergence Analysis: The paper presents a comprehensive analysis of convergence guarantees, including global exponential convergence, under less restrictive assumptions compared to existing methods. This involves weaker conditions on objective functions and constraints, expanding the applicability of the proposed dynamics to real-world problems.
3. Numerical Experiments: Computational experiments demonstrate the practicality and effectiveness of the proposed dynamics in applications ranging from signal processing to machine learning and distributed optimization. These experiments support the theoretical findings on global convergence properties and underline the benefits of the approach in distributed settings.
4. Implications for Distributed Computing: The research showcases the suitability of these dynamics for distributed computing environments. In particular, it addresses the challenges encountered in consensus optimization problems often seen in networked systems, providing a natural fit for parallel processing architectures.

Implications and Future Work

The theoretical advancements outlined in this paper have significant implications for both the development of optimization algorithms and their application to distributed systems. By establishing stability and convergence under less restrictive conditions, this research paves the way for more robust and efficient solutions to complex optimization problems.

In future work, extending these dynamics to non-convex settings and exploring acceleration techniques could further enhance their applicability and performance. Additionally, integrating machine learning-based adaptations could offer adaptive methods that optimize for varying conditions and constraints dynamically encountered in real-world environments.

Overall, this paper contributes meaningfully to the field of optimization by proposing a well-justified and theoretically sound approach, addressing both the convergence and practical implementation challenges associated with multi-block optimization problems. It opens avenues for future exploration, potentially revolutionizing how composite optimization tasks are approached in distributed and large-scale problems.