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

Abstract

This paper studies the exponential stability of the 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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</sub> , <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</sub> 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. Two quadratic Lyapunov functions are designed to prove the exponential stability. Lastly, numerical experiments further complement the theoretical analysis.