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Type of Document Dissertation Author Qi, Chunhong Author's Email Address cq05@fsu.edu URN etd-04202011-170508 Title Numerical Optimization Methods On Riemannian Manifolds Degree Doctor of Philosophy Department Mathematics, Department of Advisory Committee
Advisor Name Title Kyle A. Gallivan Committee Chair Pierre-Antoine Absil Committee Co-Chair Giray Okten Committee Member Gordon Erlebacher Committee Member M. Yousuff Hussaini Committee Member Dennis Duke University Representative Keywords
- Riemannian Manifolds
- Vector Transport
- Retraction
- Riemannian BFGS Method
- Riemannian ARC Method
- Convergence Analysis
Date of Defense 2011-03-22 Availability unrestricted Abstract This dissertation considers the generalization of two well-known unconstrainedoptimization 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.
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