An Adaptive Proximal Point Method for Nonsmooth and Nonconvex Optimization on Hadamard Manifolds

This paper addresses a class of nonsmooth and nonconvex optimization problems defined on complete Riemannian manifolds. The objective function has a composite structure, combining convex, differentiable, and lower semicontinuous terms, thereby generalizing the classical framework of difference-of-convex programming. Motivated by recent advances in proximal point methods in Euclidean and Riemannian settings, we propose two variants: one that uses the Lipschitz constant of the gradient of the smooth part, suitable when this parameter is accessible, and another that dispenses with such knowledge, expanding its applicability. We analyze the complexity of both approaches, establish their convergence, and illustrate their effectiveness through numerical experiments.