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The role of strong convexity-concavity in the convergence and robustness of the saddle-point dynamics

Cherukuri, Ashish and Mallada, Enrique and Low, Steven and Cortés, Jorge (2016) The role of strong convexity-concavity in the convergence and robustness of the saddle-point dynamics. In: 54th Annual Allerton Conference on Communication, Control, and Computing. IEEE , Piscataway, NJ, pp. 504-510. ISBN 978-1-5090-4550-1. http://resolver.caltech.edu/CaltechAUTHORS:20170217-154230896

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Abstract

This paper studies the projected saddle-point dynamics for a twice differentiable convex-concave function, which we term saddle function. The dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and (projected) gradient ascent in variables corresponding to concavity. We provide a novel characterization of the omega-limit set of the trajectories of these dynamics in terms of the diagonal Hessian blocks of the saddle function. Using this characterization, we establish global asymptotic convergence of the dynamics under local strong convexity-concavity of the saddle function. If this property is global, and for the case when the saddle function takes the form of the Lagrangian of an equality constrained optimization problem, we establish the input-to-state stability of the saddle-point dynamics by providing an ISS Lyapunov function. Various examples illustrate our results.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1109/ALLERTON.2016.7852273DOIArticle
http://resolver.caltech.edu/CaltechAUTHORS:20171204-104144520Related ItemJournal Article
ORCID:
AuthorORCID
Mallada, Enrique0000-0003-1568-1833
Low, Steven0000-0001-6476-3048
Additional Information:© 2016 IEEE. We would like to thank Simon K. Niederländer for discussions on Lyapunov functions for the saddle-point dynamics. This work was supported by NSF award ECCS-1307176 and ARPA-e Cooperative Agreement DE-AR0000695 (AC and JC), NSF CPS grant CNS 1544771 (EM), and NSF CNS grant 1545096 (SL).
Funders:
Funding AgencyGrant Number
NSFECCS-1307176
Department of Energy (DOE)DE-AR0000695
NSFCNS-1544771
NSFCNS-1545096
Record Number:CaltechAUTHORS:20170217-154230896
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170217-154230896
Official Citation:A. Cherukuri, E. Mallada, S. Low and J. Cortés, "The role of strong convexity-concavity in the convergence and robustness of the saddle-point dynamics," 2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, IL, USA, 2016, pp. 504-510. doi: 10.1109/ALLERTON.2016.7852273 URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=7852273&isnumber=7852197
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:74410
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:18 Feb 2017 03:50
Last Modified:31 Jul 2018 16:58

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