On the Exponential Stability of Primal-Dual Gradient Dynamics

Abstract

Continuous time primal-dual gradient dynamics that find a saddle point of a Lagrangian of an optimization problem have been widely used in systems and control. While the global asymptotic stability of such dynamics has been well-studied, it is less studied whether they are globally exponentially stable. In this paper, we study the primal-dual gradient dynamics for convex optimization with strongly-convex and smooth objectives and affine equality or inequality constraints, and prove global exponential stability for such dynamics. Bounds on decaying rates are provided.

arXiv commentSingle column, 24 pages, 2 figures. Shorter version accepted to L-CSS. Version 3: added Appendix I and J which extends the exponential stability results for affine inequality constraint case by relaxing the rank assumption (Assumption 2) to a weaker assumption (Assumption 4). Version 2: added Appendix-G and Appendix-H on two sided inequality constraints

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• Design and Stability of Load-Side Primary Frequency Control in Power Systems
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