Interior-point methods offer a highly versatile framework for convex optimization that is effective in theory and practice. A key notion in their theory is that of a self-concordant barrier. We give a suitable generalization of self-concordance to Riemannian manifolds and show that it gives the same structural results and guarantees as in the Euclidean setting, in particular local quadratic convergence of Newton's method. We then analyze a short-step path-following method for optimizing compatible objectives over a convex domain for which one has a self-concordant barrier, and obtain the standard complexity guarantees as in the Euclidean setting. We show that on the positive-definite matrices and other symmetric spaces, the squared distance to a point is a self-concordant function. Our work is motivated by recent progress on scaling problems and non-commutative optimization, and we show that these fit into our framework, yielding algorithms with state-of-the-art complexity guarantees. Furthermore, we show how to apply our methods to computing geometric medians on spaces with constant negative curvature.

翻译：内点方法为凸优化提供了一个高度通用的框架，在理论和实践中均有效。其理论中的一个关键概念是自和谐障碍函数。我们给出了黎曼流形上自和谐性的适当推广，并证明其与欧几里得环境中的结构结果和保证相同，特别是牛顿法的局部二次收敛性。随后，我们分析了一种短步路径跟踪方法，用于在具有自和谐障碍函数的凸域上优化兼容目标，并获得了与欧几里得环境中相同的标准复杂度保证。我们证明，在正定矩阵和其他对称空间上，到一个点的平方距离是自和谐函数。我们的工作受缩放问题和非交换优化近期进展的启发，并证明这些问题符合我们的框架，从而得到具有最先进复杂度保证的算法。此外，我们展示了如何将方法应用于计算常负曲率空间上的几何中位数。