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In this paper, we study the stability and convergence of continuous-timeLagrangian saddle flows to solutions of a convex constrained optimizationproblem. Convergence of these flows is well-known when the underlying saddlefunction is either strictly convex in the primal or strictly concave in thedual variables. In this paper, we show convergence under non-strict convexitywhen a simple, unilateral augmentation term is added. For this purpose, weestablish a novel, non-trivial characterization of the limit set of saddle-flowtrajectories that allows us to preclude limit cycles. With our presentation wetry to unify several existing problem formulations as a projected dynamicalsystem that allows projection of both the primal and dual variables, thuscomplementing results available in the recent literature.

Limit Behavior and the Role of Augmentation in Projected Saddle Flows for Convex Optimization