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 R ⁿ (corresponding to constrained optimization) and the R ⁿ 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 R ⁿ , the new algorithms are very efficient. Convergence rates are obtained. Applications are discussed. This paper is based on part of the Ph.D Thesis of the author under the supervision of Professor Tits, University of Maryland, College Park, Maryland. The author is in debt to him for invaluable suggestions on earlier versions of this paper. The author is grateful to the Associate Editor and anonymous reviewers, who pointed out a number of papers that have been included in the references; they made also detailed suggestions that lead to significant improvements of the paper. Finally, the author thanks Dr. S.T. Smith for making available his Ph.D Thesis.