Collective Optimization on Riemannian Manifolds with Bounded Curvature

Abstract: In this paper, we develop an intrinsic consensus-based optimization framework on Riemannian manifolds with bounded sectional curvature. In contrast to extrinsic approaches based on an ambient Euclidean embedding, our model is formulated directly in terms of the Riemannian structure, using logarithmic and exponential maps induced by the intrinsic geodesic distance. We prove the global well-posedness of the proposed particle system and its associated McKean--Vlasov dynamics. We also establish the global convergence of the mean-field equation toward a global minimizer of the objective function under suitable conditions. Numerical experiments on the sphere, hyperbolic space, and the special orthogonal group demonstrate the effectiveness of the intrinsic CBO dynamics for nonconvex optimization problems on manifolds.