On the Regularization of Primal-Dual Dynamics in Time-Dependent Nonconvex Optimization

Abstract

We study continuous-time primal-dual dynamics for tracking solutions of time-dependent constrained optimization problems. First, for uniformly strongly convex-strongly concave saddle functions, we establish explicit finite-time tracking and additive-disturbance bounds. For convex problems, we then introduce a directional primal regularization, combined with dual Tikhonov regularization, and quantify the resulting optimization bias and constraint violation. For smooth nonconvex problems, we couple a proximal local convexification with a two-time-scale primal-dual flow and obtain a local tracking result under constraint qualification, second-order sufficiency, and temporal regularity. Numerical traffic-signal experiments at a single intersection evaluate the proposed method under vehicle-throughput and pedestrian-aware objectives, comparing it with standard signal-control schemes.