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 Hadamard manifolds. The objective function has a composite structure involving convex, differentiable, and lower semicontinuous terms. This formulation is generally nonconvex and contains the classical difference-of-convex (DC) framework as a particular case. The generalization lies in allowing more general composite decompositions, beyond the difference of two convex functions, in the Hadamard manifold setting. 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 does not require such knowledge, thereby broadening its applicability. We analyze the complexity of both approaches, establish their convergence properties, and illustrate their effectiveness through numerical experiments.