# Fixed-Time Stable Gradient Flows: Applications to Continuous-Time Optimization

arXiv: https://arxiv.org/abs/1808.10474 · HTML: https://mlsanity.com/paper/1808.10474/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Aug 2018.

## Built on by (most cited first, 10 of 10)

- [Differentially Private Gradient Flow based on the Sliced Wasserstein Distance](https://mlsanity.com/paper/2312.08227/) (2023)
- [SGD as Free Energy Minimization: A Thermodynamic View on Neural Network Training](https://mlsanity.com/paper/2505.23489/) (2025)
- [Breaking the Convergence Barrier: Optimization via Fixed-Time Convergent Flows](https://mlsanity.com/paper/2112.01363/) (2021)
- [Learning Gradient Flow: Using Equation Discovery to Accelerate Engineering Optimization](https://mlsanity.com/paper/2602.13513/) (2026)
- [Distributed Finite-Sum Constrained Optimization subject to Nonlinearity on the Node Dynamics](https://mlsanity.com/paper/2203.14527/) (2022)
- [Fixed-Time Convergence for a Class of Nonconvex-Nonconcave Min-Max Problems](https://mlsanity.com/paper/2207.12845/) (2022)
- [Generalized Gradient Flows with Provable Fixed-Time Convergence and Fast Evasion of Non-Degenerate Saddle Points](https://mlsanity.com/paper/2212.03765/) (2022)
- [Accelerated Optimization Landscape of Linear-Quadratic Regulator](https://mlsanity.com/paper/2307.03590/) (2023)
- [From exponential to finite/fixed-time stability: Applications to optimization](https://mlsanity.com/paper/2409.11713/) (2024)
- [An Invariant Compiler for Neural ODEs in AI-Accelerated Scientific Simulation](https://mlsanity.com/paper/2603.23861/) (2026)

## Abstract (arXiv)

This paper proposes novel gradient-flow schemes that yield convergence to the optimal point of a convex optimization problem within a textitfixed time from any given initial condition for unconstrained optimization, constrained optimization, and min-max problems. The application of the modified gradient flow to unconstrained optimization problems is studied under the assumption of gradient-dominance. Then, a modified Newton's method is presented that exhibits fixed-time convergence under some mild conditions on the objective function. Building upon this method, a novel technique for solving convex optimization problems with linear equality constraints that yields convergence to the optimal point in fixed time is developed. More specifically, constrained optimization problems formulated as min-max problems are considered, and a novel method for computing the optimal solution in fixed-time is proposed using the Lagrangian dual. Finally, the general min-max problem is considered, and a modified scheme to obtain the optimal solution of saddle-point dynamics in fixed time is developed. Numerical illustrations that compare the performance of the proposed method against Newton's method, rescaled-gradient method, and Nesterov's accelerated method are included to corroborate the efficacy and applicability of the modified gradient flows in constrained and unconstrained optimization problems.