Saddle-Point Dynamics: Conditions for Asymptotic Stability of Saddle Points
Abstract
This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient descent in the first variable and gradient ascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics, the function along the proximal normals to the saddle set, and the linearity of the function in one variable. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.
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Information & Authors
Information
Published In
SIAM Journal on Control and Optimization
Pages: 486 - 511
DOI: 10.1137/15M1026924
ISSN (online): 1095-7138
Copyright
© 2017, Society for Industrial and Applied Mathematics.
History
Submitted: 19 June 2015
Accepted: 18 October 2016
Published online: 22 February 2017
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Natural Sciences and Engineering Research Council of Canada http://dx.doi.org/10.13039/501100000038
Division of Electrical, Communications and Cyber Systems http://dx.doi.org/10.13039/100000148 : 1307176
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