Globally Convergent Optimization Algorithms on Riemannian Manifolds: Uniform Framework for Unconstrained and Constrained Optimization.

This paper proposes several globally convergent geometric optimization algorithms on Riemannian manifolds, which extend some existing geometric optimization techniques. Since any set of smooth constraints in the Euclidean space Rⁿ (corresponding to constrained optimization) and the Rⁿ space itself (corresponding to unconstrained optimization) are both special Riemannian manifolds, and since these algorithms are developed on general Riemannian manifolds, the techniques discussed in this paper provide a uniform framework for constrained and unconstrained optimization problems. Unlike some earlier works, the new algorithms have less restrictions in both convergence results and in practice. For example, global minimization in the one-dimensional search is not required. All the algorithms addressed in this paper are globally convergent. For some special Riemannian manifold other than Rⁿ, the newalgorithms are very efficient. Convergence rates are obtained. Applications are discussed. [ABSTRACT FROM AUTHOR]