Numerical Optimization Methods on Riemannian Manifolds

This dissertation considers the generalization of two well-known unconstrained optimization algorithms for Rn to solve optimization problems whose constraints can be characterized as a Riemannian manifold. Efficiency and effectiveness are obtained compared to more traditional approaches to Riemannian optimization by applying the concepts of retraction and vector transport. We present a theory of building vector transports on submanifolds of Rn and use the theory to assess convergence conditions and computational efficiency of the Riemannian optimization algorithms. We generalize the BFGS method which is an highly effective quasi-Newton method for unconstrained optimization on Rn. The Riemannian version, RBFGS, is developed and its convergence and efficiency analyzed. Conditions that ensure superlinear convergence are given. We also consider the Euclidean Adaptive Regularization using Cubics method (ARC) for unconstrained optimization on Rn. ARC is similar to trust region methods in that it uses a local model to determine the modification to the current estimate of the optimal solution. Rather than a quadratic local model and constraints as in a trust region method, ARC uses a parameterized local cubic model. We present a generalization, the Riemannian Adaptive Regularization using Cubics method (RARC), along with global and local convergence theory. The efficiency and effectiveness of the RARC and RBFGS methods are investigated and their performance compared to the predictions made by the convergence theory via a series of optimization problems on various manifolds.