Advances in Computational Mathematics

, Volume 42, Issue 2, pp 333–360 | Cite as

ε-subgradient algorithms for locally lipschitz functions on Riemannian manifolds



This paper presents a descent direction method for finding extrema of locally Lipschitz functions defined on Riemannian manifolds. To this end we define a set-valued mapping \(x\rightarrow \partial _{\varepsilon } f(x)\) named ε-subdifferential which is an approximation for the Clarke subdifferential and which generalizes the Goldstein- ε-subdifferential to the Riemannian setting. Using this notion we construct a steepest descent method where the descent directions are computed by a computable approximation of the ε-subdifferential. We establish the global convergence of our algorithm to a stationary point. Numerical experiments illustrate our results.


Riemannian manifolds Lipschitz function Descent direction Clarke subdifferential 

Mathematics Subject Classifications (2010)

49J52 65K05 58C05 


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.ETH Zürich, Seminar for Applied MathematicsZürichSwitzerland

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