From exponential to finite/fixed-time stability: Applications to optimization
This provides a general method for accelerating optimization algorithms to achieve convergence in finite time, which is incremental but useful for practitioners in optimization.
The paper addresses the lack of a unified framework for converting exponentially stable optimization algorithms into finite/fixed-time stable ones, showing that this can be achieved through a simple scaling of the dynamics and verified using existing Lyapunov functions.
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.