# From exponential to finite/fixed-time stability: Applications to optimization

arXiv: https://arxiv.org/abs/2409.11713 · HTML: https://mlsanity.com/paper/2409.11713/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 40 references that are in the map.
- First posted on arXiv 18 Sep 2024.

## Builds on (most cited first, 17 of 17)

- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Gradient Descent Can Take Exponential Time to Escape Saddle Points](https://mlsanity.com/paper/1705.10412/) (2017)
- [Revisiting Normalized Gradient Descent: Fast Evasion of Saddle Points](https://mlsanity.com/paper/1711.05224/) (2017)
- [Optimization with Momentum: Dynamical, Control-Theoretic, and Symplectic Perspectives](https://mlsanity.com/paper/2002.12493/) (2020)
- [On Constraints in First-Order Optimization: A View from Non-Smooth Dynamical Systems](https://mlsanity.com/paper/2107.08225/) (2021)
- [A Hamilton-Jacobi-based Proximal Operator](https://mlsanity.com/paper/2211.12997/) (2022)
- [On the Exponential Stability of Primal-Dual Gradient Dynamics](https://mlsanity.com/paper/1803.01825/) (2018)
- [The proximal augmented Lagrangian method for nonsmooth composite optimization](https://mlsanity.com/paper/1610.04514/) (2016)
- [Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization](https://mlsanity.com/paper/1808.10474/) (2018)
- [Control Barrier Function Based Design of Gradient Flows for Constrained Nonlinear Programming](https://mlsanity.com/paper/2204.01930/) (2022)
- [A Nonsmooth Dynamical Systems Perspective on Accelerated Extensions of ADMM](https://mlsanity.com/paper/1808.04048/) (2018)
- [Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization](https://mlsanity.com/paper/1903.09580/) (2019)
- [Fixed-Time Stable Proximal Dynamical System for Solving MVIPs](https://mlsanity.com/paper/1908.03517/) (2019)
- [Stability and instability in saddle point dynamics -- Part I](https://mlsanity.com/paper/1707.07349/) (2017)
- [Time-Varying Optimization of LTI Systems via Projected Primal-Dual Gradient Flows](https://mlsanity.com/paper/2101.01799/) (2021)

## Abstract (arXiv)

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.