Exponential Stability of Primal-Dual Gradient Dynamics with Non-Strong Convexity
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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2019
- DOI:
- 10.48550/arXiv.1905.00298
- arXiv:
- arXiv:1905.00298
- Bibcode:
- 2019arXiv190500298C
- Keywords:
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- Computer Science - Systems and Control;
- Mathematics - Optimization and Control
- E-Print:
- 8 pages