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arXiv · 1905.00298

Exponential Stability of Primal-Dual Gradient Dynamics with Non-Strong Convexity

Xin Chen·Na Li
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Abstract

This paper studies the exponential stability of primal-dual gradient dynamics (PDGD) for solving convex optimization problems where constraints are in the form of Ax+By= d and the objective is min f(x)+g(y) with strongly convex smooth f but only convex smooth g. We show that when g is a quadratic function or when g and matrix B together satisfy an inequality condition, the PDGD can achieve global exponential stability given that matrix A is of full row rank. These results indicate that the PDGD is locally exponentially stable with respect to any convex smooth g under a regularity condition. To prove the exponential stability, two quadratic Lyapunov functions are designed. Lastly, numerical experiments further complement the theoretical analysis.

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BibTeXRIS

Xin Chen, Na Li. 2019-05-01. Exponential Stability of Primal-Dual Gradient Dynamics with Non-Strong Convexity. https://arxiv.org/abs/1905.00298

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