We extend the classical primal-dual interior point method from the Euclidean setting to the Riemannian one. Our method, named the Riemannian interior point method, is for solving Riemannian constrained optimization problems. We establish its local superlinear and quadratic convergence under the standard assumptions. Moreover, we show its global convergence when it is combined with a classical line search. Our method is a generalization of the classical framework of primal-dual interior point methods for nonlinear nonconvex programming. Numerical experiments show the stability and efficiency of our method.

中文翻译：

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流形约束优化的黎曼内点法

我们将经典的原对偶内点方法从欧几里得设置扩展到黎曼设置。我们的方法称为黎曼内点法，用于解决黎曼约束优化问题。我们在标准假设下建立了其局部超线性和二次收敛。此外，当它与经典的线搜索相结合时，我们展示了它的全局收敛性。我们的方法是非线性非凸规划的原始对偶内点方法的经典框架的推广。数值实验表明了我们方法的稳定性和效率。