Riemannian Conjugate Gradient Methods: General Framework and Specific Algorithms with Convergence Analyses

This paper proposes a novel general framework of Riemannian conjugate gradient methods, that is, methods on manifolds. The are important first-order optimization algorithms both in Euclidean spaces and While various types studied spaces, there have been fewer studies those In each iteration the previous search direction must be transported to current tangent space so it can added negative objective function at point. There several approaches transport vector another space. Therefore, more variants than case. order investigate them detail, proposed unifies existing such as ones utilizing or inverse retraction also develops other not covered studies. Furthermore, sufficient conditions for convergence class clarified. Moreover, global properties specific extensively analyzed. analyses provide theoretical results some setting completely new developments algorithms. Numerical experiments performed confirm validity results. compare performances framework.