将指数稳定优化算法改造为有限时间稳定，只需简单缩放动态方程右侧。
From exponential to finite/fixed-time stability: Applications to optimization
- 通过缩放原系统右侧项实现有限时间收敛
- 利用原系统的李雅普诺夫函数验证新算法稳定性
- 适用于非光滑复合问题与带线性约束的光滑问题
现有有限/固定时间稳定优化算法多针对具体问题实例，缺乏统一框架，阻碍了对复杂算法（如原始-对偶梯度流）的理解。本文回答核心问题：给定一个指数稳定的优化算法，能否将其改造为有限/固定时间稳定？我们给出肯定答案，提出仅需对原动态系统的右侧进行简单缩放，即可在有限时间内计算出解，并利用原系统证明指数稳定的李雅普诺夫函数，严格认证新算法的所需性质。最后，我们在非光滑复合优化问题和具有线性约束的光滑问题上验证了该方法的有效性。
原文摘要 · Abstract (English)
The development of finite/fixed-time stable optimization algorithms typically involves study of specific problem instances. The lack of a unified framework hinders understanding of more sophisticated algorithms, e.g., primal-dual gradient flow dynamics. The purpose of this paper is to address the following question: Given an exponentially stable optimization algorithm, can it be modified to obtain a finite/fixed-time stable algorithm? We provide an affirmative answer, demonstrate how the solution can be computed on a finite-time interval via a simple scaling of the right-hand-side of the original dynamics, and certify the desired properties of the modified algorithm using the Lyapunov function that proves exponential stability of the original system. Finally, we examine nonsmooth composite optimization problems and smooth problems with linear constraints to demonstrate the merits of our approach.
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