Gradient and Subgradient Methods in the Poincaré Space and the Space of Positive Definite Matrices

Rios Cortegana J.E.; Papa Quiroz E.A. · Revista Politecnica (2026) · DOI: 10.33333/rp.vol57n1.03

Resumen

This article focuses on two main issues within the framework of optimization on Riemannian manifolds. The first concerns the computational difficulty of explicitly implementing geodesics that originate from a point in a given tangent direction on the upper half-plane manifold of Poincaré. Although the differential equation for the geodesic associated with the Christoffel symbols can be solved analytically, the resulting expressions are too complex to be used in iterative algorithms. To overcome this limitation, we propose a numerical strategy based on the fourth-order Runge– Kutta method to approximate the geodesic and perform key computations for gradient and subgradient methods in this context.The second issue is the computational implementation of gradient and subgradient methods on the Poincaré upper half-plane and on the manifold of symmetric positive definite (SPD) matrices. In the case of minimizing non-differentiable functions, it is necessary, for instance, to compute the convex subdifferential or the Greenberg–Pierskalla subdifferential for quasi-convex functions. We will analyze these implementations in both spaces considered as Riemannian manifolds and obtain advantageous results when compared to methods implemented in the Euclidean setting.

Instituciones: Universidad Nacional Mayor de San Marcos