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

TL;DR

This paper analyzes the exponential stability of primal-dual gradient dynamics for convex optimization problems with non-strongly convex components, providing conditions under which global or local exponential stability is achieved.

Contribution

It establishes new stability results for PDGD when the objective g is convex but not strongly convex, under specific regularity and matrix conditions.

Findings

PDGD is globally exponentially stable when g is quadratic or satisfies certain inequalities.

PDGD is locally exponentially stable under regularity conditions.

Numerical experiments support the theoretical stability results.

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.