# Control Barrier Function Based Design of Gradient Flows for Constrained Nonlinear Programming

arXiv: https://arxiv.org/abs/2204.01930 · HTML: https://mlsanity.com/paper/2204.01930/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2022.

## Built on by (most cited first, 8 of 8)

- [Accelerated forward-backward and Douglas-Rachford splitting dynamics](https://mlsanity.com/paper/2407.20620/) (2024)
- [Anytime Safe Reinforcement Learning](https://mlsanity.com/paper/2504.16417/) (2025)
- [Safe Gradient Flow for Bilevel Optimization](https://mlsanity.com/paper/2501.16520/) (2025)
- [Energy-Based Dynamical Models for Neurocomputation, Learning, and Optimization](https://mlsanity.com/paper/2604.05042/) (2026)
- [From exponential to finite/fixed-time stability: Applications to optimization](https://mlsanity.com/paper/2409.11713/) (2024)
- [Sequential QCQP for Bilevel Optimization with Line Search](https://mlsanity.com/paper/2505.14647/) (2025)
- [Off-Policy Reinforcement Learning with Anytime Safety Guarantees via Robust Safe Gradient Flow](https://mlsanity.com/paper/2510.01492/) (2025)
- [A Control-Barrier-Function-Based Algorithm for Policy Adaptation in Reinforcement Learning](https://mlsanity.com/paper/2510.02720/) (2025)

## Abstract (arXiv)

This paper considers the problem of designing a continuous-time dynamical system that solves a constrained nonlinear optimization problem and makes the feasible set forward invariant and asymptotically stable. The invariance of the feasible set makes the dynamics anytime, when viewed as an algorithm, meaning it returns a feasible solution regardless of when it is terminated. Our approach augments the gradient flow of the objective function with inputs defined by the constraint functions, treats the feasible set as a safe set, and synthesizes a safe feedback controller using techniques from the theory of control barrier functions. The resulting closed-loop system, termed safe gradient flow, can be viewed as a primal-dual flow, where the state corresponds to the primal variables and the inputs correspond to the dual ones. We provide a detailed suite of conditions based on constraint qualification under which (both isolated and nonisolated) local minimizers are stable with respect to the feasible set and the whole state space. Comparisons with other continuous-time methods for optimization in a simple example illustrate the advantages of the safe gradient flow.