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Mathematics > Optimization and Control

arXiv:1812.00613 (math)
[Submitted on 3 Dec 2018]

Title:Running Primal-Dual Gradient Method for Time-Varying Nonconvex Problems

Authors:Yujie Tang, Emiliano Dall'Anese, Andrey Bernstein, Steven Low
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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.
Subjects: Optimization and Control (math.OC)
Cite as: arXiv:1812.00613 [math.OC]
  (or arXiv:1812.00613v1 [math.OC] for this version)
  https://doi.org/10.48550/arXiv.1812.00613

Submission history

From: Yujie Tang [view email]
[v1] Mon, 3 Dec 2018 09:06:42 UTC (1,916 KB)
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