Tannenbaum's Gain-Margin Optimization Meets Polyak's Heavy-Ball Algorithm

• 成果类型: Article
• 署名作者: Wu, Wuwei; Chen, Jie; Jovanovic, Mihailo R.; Georgiou, Tryphon T.
• 署名单位: City University of Hong Kong; University of Southern California; University of California System; University of California Irvine
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2026.3678501
• 发表日期: 2026
• 关键词: INTERNAL-MODEL PRINCIPLE; TIME-INVARIANT PLANTS; feedback stabilization; systems; interpolation; uncertainty; monotone

摘要

This article highlights an apparent, yet relatively unknown link between algorithm design in optimization theory and controller synthesis in robust control. Specifically, quadratic optimization can be recast as a regulation problem within the framework of H-infinity control. From this vantage point, the optimality of Polyak's fastest heavy-ball algorithm can be ascertained as a solution to a gain-margin optimization problem. The approach is independent of Polyak's original and brilliant argument, and relies on foundational work by Tannenbaum, who introduced and solved gain-margin optimization via Nevanlinna-Pick interpolation theory. The link between first-order optimization methods and robust control sheds new light on the limits of algorithmic performance of such methods, and suggests a framework where similar computational tasks can be systematically studied and algorithms optimized. In particular, it raises the question as to whether periodically scheduled algorithms can achieve faster rates for quadratic optimization, in a manner analogous to periodic control that extends the gain margin beyond that of time-invariant control. This turns out not to be the case, due to the analytic obstruction of a transmission zero that is inherent in causal schemes. Interestingly, this obstruction can be removed with implicit algorithms, cast as feedback regulation problems with causal, but not strictly causal dynamics, thereby devoid of the transmission zero at infinity and able to achieve superior convergence rates.