On the global exponential stability of primal-dual dynamics for convex problems with linear equality constraints

We examine global exponential stability of the primal-dual gradient flow dynamics for differentiable convex problems with linear equality constraints. We show that if the set of equilibrium points is affine, then, regardless of the initial conditions, trajectories of the gradient flow move in the direction that is perpendicular to equilibrium set. When the objective function is strongly convex, we utilize this structure to show that the primal-dual dynamics are globally exponentially stable even if the constraint matrix is not full-row rank. We also provide an explicit characterization of the exponential convergence rate in terms of the smallest nonzero singular value of the constraint matrix.