Sharpest contraction rate for primal–dual dynamics with linear equality constraints

Determine the sharpest exponential contraction rate for the continuous-time primal–dual dynamics that solve linear equality-constrained minimization problems; specifically, identify the largest rate and an associated norm or Riemannian metric under which the flow is strongly contracting so that the distance between any two trajectories decays exponentially.

Background

The paper highlights that while sharp characterizations of contractivity exist for certain systems (e.g., gradient flow, firing-rate neural networks, Lur’e models), for other standard dynamics the best contraction rates are not yet characterized.

Primal–dual dynamics for linear equality-constrained minimization are a canonical algorithmic flow used to solve problems of the form minimize f(x) subject to Ax=b. Establishing the largest contraction rate would yield tight robustness and convergence guarantees and unify analysis across normed and Riemannian settings.