Distributed Optimization with Globally Coupled Equality Constraint: Implicit Tracking-Based Algorithm Design and Analysis

Concetti Chiave

The core message of this work is to design efficient distributed algorithms, IDEA and Proj-IDEA, to solve a class of distributed optimization problems with a globally coupled equality constraint and local constrained sets. The key innovation is the novel implicit tracking approach, which allows the distributed algorithms to converge without the need of the strict convexity of local cost functions.

Sintesi

The paper studies a class of distributed optimization problem with a globally coupled equality constraint and local constrained sets. For the special case where local constrained sets are absent, an augmented primal-dual gradient dynamics (APGD) is proposed and analyzed. However, APGD cannot be implemented distributedly since the violation of the coupled constraint needs to be used.

To address this issue, the authors propose a novel implicit tracking approach, which is essentially different from the explicit tracking approach used in previous works. This leads to the birth of the implicit tracking-based distributed augmented primal-dual gradient dynamics (IDEA). A projected variant of IDEA, i.e., Proj-IDEA, is further designed to deal with the general case where local constrained sets exist.

The convergence of IDEA and Proj-IDEA are analyzed over undigraphs and digraphs, respectively. The key results are:

1. Under undigraphs, IDEA and Proj-IDEA can converge when local cost functions are only convex, without the need of strict or strong convexity. This is the first constant step-size distributed algorithm that can solve the studied problem in this general setting.

2. When local cost functions are strongly convex and smooth, IDEA can achieve exponential convergence with a weaker condition on the coupled constraint, compared to existing works.

3. Under digraphs, the convergence of Proj-IDEA can be guaranteed when local cost functions are strongly convex, while IDEA can converge exponentially if the local constraint matrices have a special structure.

4. The convergence analysis is based on the Lyapunov stability theory, which provides a deep understanding of the relation between APGD and IDEA, leading to the design of nice Lyapunov functions.

5. The implicit tracking approach can reduce the number of state variables that need to be exchanged, compared to the explicit tracking-based approach. Numerical experiments show that IDEA usually has a faster convergence rate than the explicit tracking-based algorithm.

Statistiche

The paper does not contain any explicit numerical data or statistics to support the key claims. The analysis is mainly theoretical, focusing on the algorithm design and convergence analysis.

Citazioni

There are no striking quotes from the content that directly support the author's key logics.

Approfondimenti chiave tratti da

Implicit Tracking-Based Distributed Constraint-Coupled Optimization

by Jingwang Li,... alle arxiv.org 04-01-2024

https://arxiv.org/pdf/2201.07627.pdf

Domande più approfondite

What are some practical applications of the studied distributed optimization problem with a globally coupled equality constraint

The studied distributed optimization problem with a globally coupled equality constraint has practical applications in various fields such as large-scale machine learning, distributed control, decentralized estimation, and smart grid management. For example, in resource allocation scenarios, the problem can help optimize the allocation of resources among multiple agents while satisfying global constraints. In economic dispatch, the problem can aid in optimizing the generation and distribution of power resources across a network. Network utility maximization can benefit from this optimization to enhance the overall efficiency and performance of communication networks by balancing various utility functions subject to global constraints.

How can the proposed algorithms be extended to handle time-varying communication graphs or asynchronous updates among agents

To extend the proposed algorithms to handle time-varying communication graphs or asynchronous updates among agents, adjustments can be made in the algorithm design to accommodate the dynamic nature of the network. For time-varying graphs, the algorithms can incorporate mechanisms to adapt to changes in the communication structure over time. This may involve updating the adjacency matrix or adjusting the communication protocols based on the evolving graph topology. Asynchronous updates among agents can be addressed by introducing synchronization mechanisms or incorporating delay-tolerant strategies to ensure convergence despite the lack of simultaneous updates.

Can the implicit tracking approach be applied to design distributed algorithms for other classes of constrained optimization problems beyond the one studied in this work

The implicit tracking approach utilized in the design of distributed algorithms for the studied constrained optimization problem can be applied to other classes of constrained optimization problems as well. By leveraging the concept of tracking the violation distributedly, algorithms can be developed to handle a wide range of constrained optimization scenarios where privacy preservation and distributed computation are crucial. This approach can be extended to problems with different types of constraints, objective functions, and communication topologies, providing a versatile framework for designing efficient distributed optimization algorithms for various applications.