黎曼流形上的Fenchel对偶理论与原始-对偶算法
Fenchel Duality Theory and A Primal-Dual Algorithm on Riemannian Manifolds
AI总结：
本文将Fenchel共轭推广到黎曼流形，建立相应对偶理论，并据此提出黎曼原始-对偶算法，证明其在Hadamard流形上收敛，数值实验显示该算法在非正曲率流形上可与Douglas-Rachford算法竞争，且在正曲率流形上也能收敛。
摘要：
本文引入了一种新的Fenchel共轭概念，将经典Fenchel共轭推广到定义在黎曼流形上的函数。我们研究了其性质，例如Fenchel-Young不等式以及利用Fenchel-Moreau定理的类比刻画凸次微分。这些Fenchel共轭的性质被用于推导黎曼原始-对偶优化算法，并在适当假设下证明其在Hadamard流形情形下的收敛性。数值结果展示了该算法的性能，其可与最近在非正曲率流形上导出的Douglas-Rachford算法相竞争。此外，我们通过数值实验表明，我们的新算法甚至在正曲率流形上也能收敛。
英文摘要：
This paper introduces a new notion of a Fenchel conjugate, which generalizes the classical Fenchel conjugation to functions defined on Riemannian manifolds. We investigate its properties, e.g.,~the Fenchel--Young inequality and the characterization of the convex subdifferential using the analogue of the Fenchel--Moreau Theorem. These properties of the Fenchel conjugate are employed to derive a Riemannian primal-dual optimization algorithm, and to prove its convergence for the case of Hadamard manifolds under appropriate assumptions. Numerical results illustrate the performance of the algorithm, which competes with the recently derived Douglas--Rachford algorithm on manifolds of nonpositive curvature. Furthermore, we show numerically that our novel algorithm even converges on manifolds of positive curvature.
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