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arxiv: 1805.05565 · v1 · pith:EYCD3KZPnew · submitted 2018-05-15 · 🧮 math.OC

A Cubic Regularized Newton's Method over Riemannian Manifolds

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keywords conditioncubicepsilonmanifoldmethodnewtonregularizedriemannian
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In this paper we present a cubic regularized Newton's method to minimize a smooth function over a Riemannian manifold. The proposed algorithm is shown to reach a second-order $\epsilon$-stationary point within $\mathcal{O}(1/\epsilon^{\frac{3}{2}})$ iterations, under the condition that the pullbacks are locally Lipschitz continuous, a condition that is shown to be satisfied if the manifold is compact. Furthermore, we present a local superlinear convergence result under some additional conditions.

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  1. A primal-dual interior point trust region method for second-order stationary points of Riemannian inequality-constrained optimization problems

    math.OC 2025-01 unverdicted novelty 8.0

    RIPTRM is the first trust-region primal-dual interior-point method for Riemannian problems with nonlinear inequalities, with global convergence to second-order stationary points under strict complementarity.