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Nonsmooth Trust Region Algorithms for Locally Lipschitz Functions on Riemannian Manifolds

by P. Grohs and S. Hosseini

(Report number 2014-20)

Abstract
This paper presents a Riemannian trust region algorithm for unconstrained optimization problems with locally Lipschitz objective functions defined on complete Riemannian manifolds. To this end we define a function \(\Phi:TM\rightarrow \Bbb{R}\) on the tangent bundle \(TM\), and at \(k\)-th iteration, using the restricted function \(\Phi|_{T_{x_k}M}\) where \(T_{x_k}M\) is the tangent space at \(x_k\), a local model function \(Q_k\) that carries both first and second order information for the locally Lipschitz objective function \(f:M\rightarrow \Bbb{R}\) on a Riemannian manifold \(M\), is defined and minimized over a trust region. We establish the global convergence of the proposed algorithm. Moreover, using the Riemannian \(\varepsilon\)-subdifferential, a suitable model function is defined. Numerical experiments illustrate our results.

Keywords: Nonsmooth Optimization, Riemannian Manifolds, Trust Region Methods

@Techreport{GH14_570,
  author = {P. Grohs and S. Hosseini},
  title = {Nonsmooth Trust Region Algorithms for Locally Lipschitz Functions on Riemannian Manifolds},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2014-20},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2014/2014-20.pdf },
  year = {2014}
}

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