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 analyze a 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 provide general constructions of barriers, and show that on the space of positive-definite matrices and other symmetric spaces, the squared distance to a point is self-concordant. To demonstrate the versatility of our framework, we give algorithms with state-of-the-art complexity guarantees for the general class of scaling and non-commutative optimization problems, which have been of much recent interest, and we provide the first algorithms for efficiently finding high-precision solutions for computing minimal enclosing balls and geometric medians in nonpositive curvature.

翻译：内点法为凸优化提供了一个高度灵活的框架，在理论和实践中均有效。其理论中的核心概念是自和谐障碍函数。我们给出了自和谐性质在黎曼流形上的适当推广，并证明它能得到与欧几里得空间中相同的结构结果和保证，特别是牛顿法的局部二次收敛性。我们分析了一种路径跟踪方法，用于在具有自和谐障碍函数的凸域上优化兼容目标函数，并获得了与欧几里得空间相同的标准复杂度保证。我们提供了障碍函数的一般构造，并证明在正定矩阵空间及其他对称空间中，到某点的平方距离是自和谐的。为展示我们框架的通用性，我们针对近期备受关注的缩放类问题与非交换优化问题，给出了具有最前沿复杂度保证的算法，并首次提供了在非正曲率空间中高效计算最小包围球和几何中位数的高精度解的算法。