Saddle-Point Convergence of Constrained Primal-Dual Dynamics

In this letter, we consider saddle-point convergence of primal-dual dynamics for inequality constrained convex optimization problems. By considering the primal-dual dynamics as the interconnection of (i) a gradient system and (ii) a nonlinear controller with incremental sector-bounded static nonlinearity, we establish asymptotic stability by invoking the classical notions of Lyapunov stability, the invariance principle, along with some regularity assumptions. For the special case of strongly convex problem with affine constraints, we prove global exponential convergence. As compared to existing techniques in the literature, the proposed approach offers simple and transparent tuning guideline for robust exponential stability of the saddle-point dynamics and allows for establishing an upper bound on the exponential decay rate.