Distributed continuous-time convex optimization on weight-balanced digraphs (1204.0304v2)

Abstract: This paper studies the continuous-time distributed optimization of a sum of convex functions over directed graphs. Contrary to what is known in the consensus literature, where the same dynamics works for both undirected and directed scenarios, we show that the consensus-based dynamics that solves the continuous-time distributed optimization problem for undirected graphs fails to converge when transcribed to the directed setting. This study sets the basis for the design of an alternative distributed dynamics which we show is guaranteed to converge, on any strongly connected weight-balanced digraph, to the set of minimizers of a sum of convex differentiable functions with globally Lipschitz gradients. Our technical approach combines notions of invariance and cocoercivity with the positive definiteness properties of graph matrices to establish the results.

• The paper identifies that traditional consensus-based dynamics fail in directed networks, highlighting the need for alternative convergent methods.
• The paper introduces a new distributed dynamics with a design parameter that guarantees convergence for convex functions with Lipschitz gradients.
• The approach leverages invariance, cocoercivity, and graph matrix properties to robustly extend continuous-time optimization to weight-balanced digraphs.

An Examination of Distributed Continuous-Time Convex Optimization on Weight-Balanced Digraphs

This paper, authored by Bahman Gharesifard and Jorge Cortés, presents a comprehensive paper on distributed continuous-time optimization of a sum of convex functions over directed graphs. The authors confront the discrepancy between consensus-based dynamics used for undirected graphs and their failure to converge in the directed setting, thus laying groundwork for an alternative distributed approach guaranteed to achieve convergence.

Key Contributions

The paper makes several noteworthy contributions to the field of distributed optimization:

1. Consensus-Based Dynamics Limitation: The paper identifies that traditional consensus-based dynamics, effective in undirected graph scenarios, do not ensure convergence in directed networks. This challenge underscores the complexity added by unidirectional information flow inherent in strongly connected weight-balanced digraphs.
2. New Dynamics Proposal: To address the convergence issue on directed graphs, the paper proposes an innovative distributed dynamics approach with a design parameter. This new mechanism guarantees convergence to the minimizers of a sum of convex, differentiable functions with globally Lipschitz gradients across any strong connectivity in weight-balanced digraphs.
3. Technical Approach: The research employs a technical strategy using invariance and cocoercivity principles combined with graph matrix positive definiteness properties. This blend of mathematical theory ensures that the proposed dynamics adhere to desired convergence characteristics.
4. Proof Technique: Employing a novel proof structure, the authors show the asymptotic correctness of the saddle-point dynamics, deviating from typical Hessian-based approaches. This method adds robustness to their theoretical framework, allowing a broader application even when convexity could vary in nature.

Implications and Future Work

The implications of this research are both practical and theoretical. Practically, the insights can be applied to areas like sensor networks, source localization, and robust estimation where directed graph dynamics are prevalent. Theoretically, it advances the understanding of distributed algorithms in optimization, bridging gaps between discrete-time consensus designs and their continuous-time counterparts.

The paper suggests future directions for extending their results. These include:

• Extending convergence guarantees to locally Lipschitz functions on weight-balanced directed graphs.
• Exploring dynamics on general digraphs.
• Incorporating additional local and global constraints.
• Optimizing the design parameter in distributed environments.

Conclusion

The paper's thorough examination of distributed optimization over directed graphs not only clarifies existing challenges but also proposes viable solutions, extending the body of knowledge in distributed consensus and optimization. Through its novel techniques, it sets the stage for further exploration into dynamic, distributed, and constrained optimization scenarios in various networked systems.