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

TL;DR

This paper investigates the stability and convergence of projected saddle flows in convex optimization, demonstrating that simple augmentation ensures convergence even without strict convexity, and unifies various formulations as projected dynamical systems.

Contribution

It introduces a novel convergence result for saddle flows with unilateral augmentation under non-strict convexity, unifying multiple problem formulations as projected dynamical systems.

Findings

Convergence of saddle flows is achieved with unilateral augmentation.

A new characterization of the limit set prevents limit cycles.

Unified framework for primal-dual projected dynamical systems.

Abstract

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.