Exponential Convergence of Primal-Dual Dynamics Under General Conditions and Its Application to Distributed Optimization.

In this article, we establish the local and global exponential convergence of a primal-dual dynamics (PDD) for solving equality-constrained optimization problems without strong convexity and full row rank assumption on the equality constraint matrix. Under the metric subregularity of Karush-Kuhn-Tucker (KKT) mapping, we prove the local exponential convergence of the dynamics. Moreover, we establish the global exponential convergence of the dynamics in an invariant subspace under a technically designed condition which is weaker than strong convexity. As an application, the obtained theoretical results are used to show the exponential convergence of several existing state-of-the-art primal-dual algorithms for solving distributed optimization without strong convexity. Finally, we provide some experiments to demonstrate the effectiveness of our results.

Citation

Luyao Guo, Xinli Shi, Jinde Cao, Zihao Wang. Exponential Convergence of Primal-Dual Dynamics Under General Conditions and Its Application to Distributed Optimization. IEEE transactions on neural networks and learning systems. 2024 Apr;35(4):5551-5565

PMID: 36178998

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