We introduce the convex bundle method to solve convex, non-smooth optimization problems on Riemannian manifolds of bounded sectional curvature. Each step of our method is based on a model that involves the convex hull of previously collected subgradients, parallelly transported into the current serious iterate. This approach generalizes the dual form of classical bundle subproblems in Euclidean space. We prove that, under mild conditions, the convex bundle method converges to a minimizer. Several numerical examples implemented using Manopt.jl illustrate the performance of the proposed method and compare it to the subgradient method, the cyclic proximal point algorithm, as well as the proximal bundle method.

翻译：本文提出凸束方法，用于求解有界截面曲率黎曼流形上的凸非光滑优化问题。该方法的每一步均基于一个模型，该模型涉及将先前收集的次梯度经平行移动至当前严肃迭代点后形成的凸包。此方法推广了欧几里得空间中经典束子问题的对偶形式。我们证明在温和条件下，凸束方法收敛于极小值点。通过Manopt.jl实现的若干数值算例展示了所提方法的性能，并与次梯度法、循环邻近点算法以及邻近束方法进行了比较。