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This paper studies the exponential stability of primal-dual gradient dynamics(PDGD) for solving convex optimization problems where constraints are in theform of Ax+By= d and the objective is min f(x)+g(y) with strongly convex smoothf but only convex smooth g. We show that when g is a quadratic function or wheng and matrix B together satisfy an inequality condition, the PDGD can achieveglobal exponential stability given that matrix A is of full row rank. Theseresults indicate that the PDGD is locally exponentially stable with respect toany convex smooth g under a regularity condition. To prove the exponentialstability, two quadratic Lyapunov functions are designed. Lastly, numericalexperiments further complement the theoretical analysis.

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