Exponential Convergence of Augmented Primal-dual Gradient Algorithms for Partially Strongly Convex Functions
Mengmou Li, Masaaki Nagahara
TL;DR
The paper addresses achieving global exponential convergence for augmented primal-dual gradient methods under partial strong convexity, requiring $f$ to be $\mu$-strongly convex only on the subspace defined by equality constraints while maintaining an $l$-Lipschitz gradient. It develops an IQC/KYP-based, frequency-domain analysis that covers both fully-ranked and rank-deficient cases, and demonstrates applicability to distributed optimization via consensus constraints. Key contributions include (i) proving $\rho$-exponential convergence under partial strong convexity, (ii) unifying centralized and distributed results through IQC methods and coordinate transformations, and (iii) relaxing global convexity to RSI in the distributed setting. The results broaden exponential-convergence guarantees to problems lacking full strong convexity and inform design of distributed optimization algorithms under weaker curvature assumptions.
Abstract
We show that the augmented primal-dual gradient algorithms can achieve global exponential convergence with partially strongly convex functions. In particular, the objective function only needs to be strongly convex in the subspace satisfying the equality constraint and can be generally convex elsewhere, provided the global Lipschitz condition for the gradient is satisfied. This condition implies that states outside the equality subspace will converge towards it exponentially fast. The analysis is then applied to distributed optimization, where the partially strong convexity can be relaxed to the restricted secant inequality condition, which is not necessarily convex. This work unifies global exponential convergence results for some existing centralized and distributed algorithms.