# Running Primal-Dual Gradient Method for Time-Varying Nonconvex Problems

arXiv: https://arxiv.org/abs/1812.00613 · HTML: https://mlsanity.com/paper/1812.00613/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Dec 2018.

## Built on by (most cited first, 4 of 4)

- [Prediction techniques for dynamic imaging with online primal-dual methods](https://mlsanity.com/paper/2405.02497/) (2024)
- [Distributed Online Bandit Nonconvex Optimization with One-Point Residual Feedback via Dynamic Regret](https://mlsanity.com/paper/2409.15680/) (2024)
- [Online optimisation for dynamic electrical impedance tomography](https://mlsanity.com/paper/2412.12944/) (2024)
- [A Data-Driven Real-Time Optimal Power Flow Algorithm Using Local Feedback](https://mlsanity.com/paper/2502.15306/) (2025)

## Abstract (arXiv)

This paper considers a nonconvex optimization problem that evolves over time, and addresses the synthesis and analysis of regularized primal-dual gradient methods to track a Karush-Kuhn-Tucker (KKT) trajectory. The proposed regularized primal-dual gradient methods are implemented in a running fashion, in the sense that the underlying optimization problem changes during the iterations of the algorithms. For a problem with twice continuously differentiable cost and constraints, and under a generalization of the Mangasarian-Fromovitz constraint qualification, sufficient conditions are derived for the running algorithm to track a KKT trajectory. Further, asymptotic bounds for the tracking error (as a function of the time-variability of a KKT trajectory) are obtained. A continuous-time version of the algorithm, framed as a system of differential inclusions, is also considered and analytical convergence results are derived. For the continuous-time setting, a set of sufficient conditions for the KKT trajectories not to bifurcate or merge is proposed. Illustrative numerical results inspired by a real-world application are provided.