Nonsmooth trust region algorithms for locally Lipschitz functions on Riemannian manifolds
Grohs, P. ; Hosseini, S.
In: Ima Journal of Numerical Analysis, 2016, vol. 36, no. 3, p. 1167-1192
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- 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 $\Phi :TM\rightarrow \mathbb {R}$ on the tangent bundle $TM$, and at the $k$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 $f:M\rightarrow \mathbb {R}$ 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.