Mathematical Programming

, Volume 176, Issue 1–2, pp 403–427 | Cite as

Behavior of accelerated gradient methods near critical points of nonconvex functions

  • Michael O’Neill
  • Stephen J. WrightEmail author
Full Length Paper Series B


We examine the behavior of accelerated gradient methods in smooth nonconvex unconstrained optimization, focusing in particular on their behavior near strict saddle points. Accelerated methods are iterative methods that typically step along a direction that is a linear combination of the previous step and the gradient of the function evaluated at a point at or near the current iterate. (The previous step encodes gradient information from earlier stages in the iterative process). We show by means of the stable manifold theorem that the heavy-ball method is unlikely to converge to strict saddle points, which are points at which the gradient of the objective is zero but the Hessian has at least one negative eigenvalue. We then examine the behavior of the heavy-ball method and other accelerated gradient methods in the vicinity of a strict saddle point of a nonconvex quadratic function, showing that both methods can diverge from this point more rapidly than the steepest-descent method.


Accelerated gradient methods Nonconvex optimization 

Mathematics Subject Classification

90C26 49M30 



We are grateful to Bin Hu for his advice and suggestions on the manuscript. We are also grateful to the referees and editor for helpful suggestions.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA

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