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.
A Cubic Regularized Newton's Method over Riemannian Manifolds
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abstract
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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A primal-dual interior point trust region method for second-order stationary points of Riemannian inequality-constrained optimization problems
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.