A Busemann Hybrid Projection-Proximal Point Algorithm for Optimization Problems on Hadamard Manifolds

Abstract We study convex optimization problems on Hadamard manifolds and propose a projection based variant of the proximal point algorithm, called the Busemann hybrid projection-proximal point algorithm . The method replaces Euclidean hyperplanes by horospheres defined through Busemann functions and uses the associated projection geometry to build an intrinsic update rule. The projection step is available in closed form and avoids tangent space linearization. The method allows inexactness in the subproblem solution under a relative error level strictly below one. We establish a Fejér type descent property, prove global convergence, and derive a sublinear complexity bound. We also show that, in the exact case, the method reduces to the classical Riemannian proximal point algorithm. The results highlight the role of Busemann based support inequalities and subdifferentials in optimization on spaces of nonpositive curvature.