  In this paper, we study the stability and convergence of continuous-time
Lagrangian saddle flows to solutions of a convex constrained optimization
problem. Convergence of these flows is well-known when the underlying saddle
function is either strictly convex in the primal or strictly concave in the
dual variables. In this paper, we show convergence under non-strict convexity
when a simple, unilateral augmentation term is added. For this purpose, we
establish a novel, non-trivial characterization of the limit set of saddle-flow
trajectories that allows us to preclude limit cycles. With our presentation we
try to unify several existing problem formulations as a projected dynamical
system that allows projection of both the primal and dual variables, thus
complementing results available in the recent literature.
