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$\varepsilon$-Subgradient Algorithms for Locally Lipschitz Functions on Riemannian Manifolds

by P. Grohs and S. Hosseini

(Report number 2013-49)

Abstract
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 \(\varepsilon\)-subdifferential which is an approximation for the Clarke subdifferential and which generalizes the Goldstein-\(\varepsilon\)-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 \(\varepsilon\)-subdifferential. We establish the convergence of our algorithm to a stationary point. Numerical experiments illustrate our results.

Keywords: Riemannian manifolds, Lipschitz function, Descent direction, Clarke subdifferential.

@Techreport{GH13_549,
  author = {P. Grohs and S. Hosseini},
  title = {$\varepsilon$-Subgradient Algorithms for Locally Lipschitz Functions on Riemannian Manifolds},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-49},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-49.pdf },
  year = {2013}
}

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