Exponential Stability of Primal-Dual Gradient Dynamics with Non-Strong Convexity
arXiv:1905.00298
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.
8 pages
Cited by in corpus (4)
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- Distributed and time-varying primal-dual dynamics via contraction analysis
- Global exponential stability of primal-dual gradient flow dynamics based on the proximal augmented Lagrangian: A Lyapunov-based approach
- Exponential Stability of Partial Primal-Dual Gradient Dynamics with Nonsmooth Objective Functions