Running Primal-Dual Gradient Method for Time-Varying Nonconvex Problems
Abstract
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.
Cite
@article{arxiv.1812.00613,
title = {Running Primal-Dual Gradient Method for Time-Varying Nonconvex Problems},
author = {Yujie Tang and Emiliano Dall'Anese and Andrey Bernstein and Steven Low},
journal= {arXiv preprint arXiv:1812.00613},
year = {2018}
}