Below are some links relating to the RTR and to Riemannian Optimization in general. Feel free to email me if you have links that should appear here...
Other Riemannian Optimization Software
SG_Min - Stiefel Grassmann
Optimization software
Provides four iterations for optimization on the Steifel and Grassmann manifolds:
dog-leg steps
Polak-Ribiere nonlinear CG
Fletcher-Reeves nonlinear CG
Newton's method
Contains applications for nearest degenerate eigenproblems, least square
diagonalization problems, Procrustes problems, and electronic structures problems. Language: MATLAB Author: Ross Lippert
The following lists are intended to be a jumping off point for those looking for
resources on Riemannian optimization. However, they are grossly incomplete.
Please email me when you find neglected works.
Riemannian Optimization Books
Optimization and Dynamical Systems - Helmke and Moore. 1994, Springer-Verlag.
Convex functions and optimization methods on Riemannian manifolds - C. Udriste. Kluwer Academic Publishers, 1994.
Riemannian Optimization Articles
P.-A. Absil, C. G. Baker, and K. A. Gallivan. A truncated-CG style method for symmetric generalized eigenvalue problems J. Comput. Appl. Math. 189(1-2) (2006) 274-285.
P.-A. Absil, R. Mahony and R. Sepulchre. Riemannian geometry of Grassmann manifolds with a view on algorithmic computation. Acta Appl. Math. 80(2) (2004) 199-220.
P.-A. Absil, C. G. Baker and K. A. Gallivan. Trust-region methods on Riemannian manifolds. Foundations of Computational Mathematics 7(3) (2007) 303-330.
R. L. Adler, J.-P. Dedieu, J. Y. Margulies, M. Martens and M. Shub. Newton's method on Riemannian manifolds and a geometric model for the human spine. IMA J. Numer. Anal. 22(3) (2002) 359-390.
C. G. Baker, P.-A. Absil and K. A. Gallivan. An implicit Riemannian trust-region method for the symmetric generalized eigenproblem. Computational Science - ICCS 2006. Lecture Notes in Computer Science 3991. (2006) 210-217.
M. Beko, J. Xavier and V.A.N. Barroso. Noncoherent Communication in Multiple-Antenna Systems: Receiver Design and Codebook Construction IEEE Trans. on Signal Processing. 55(12) (2007) 5703-5715.
M. T. Chu. Numerical methods for inverse singular value problems. SIAM Journal on Numerical Analysis 29(3) (1992) 885.
M. T. Chu. A list of matrix flows with applications. Hamiltonian and gradient flows, algorithms and control, Fields Inst. Commun. 3. (1994) 87-97.
J.-P. Dedieu and D. Novitsky. Symplectic methods for the approximation of the exponential and the Newton sequence on Riemannian submanifolds. Journal of Complexity 21 (2005) 487-501.
J.-P. Dedieu, P. Priouret and G. Malajovich. Newton's method on Riemannian manifolds: convariant alpha theory. IMA J. Numer. Anal. 23(3) (2003) 395-419.
J. Dehaene and J. Vandewalle. New Lyapunov functions for the continuous-time QR algorithm. Proceedings of the 14th International Symposium on the Mathematical Theory of Networks and Systems (2000)
A. Edelman, T. A. Arias and S. T. Smith. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl. 20(2) (1998) 303-353.
D. Gabay. Minimizing a differentiable function over a differential manifold. Journal of Optimization Theory and Applications 37(2) (1982) 177-219.
K. Hüper and J. Trumpf. Newton-like methods for numerical optimization on manifolds. Proc. 38th IEEE Asilomar Conference on Signals Systems and Computers (2004).
R. Lippert and A. Edelman. Nonlinear eigenvalue problems with orthogonality constraints. Templates for the Solution of Algebraic Eigenvalue Problems, SIAM. (2000) 290-314.
E. Lundström and L. Eldén. Adaptive eigenvalue computations using Newton's method on the Grassmann manifold. SIAM J. Matrix Anal. Appl. 23(3) (2002) 819-839.
D. G. Luenberger. The gradient projection method along geodesics. Management Sci. 18 (1972) 620-631.
R. Mahony and J. H. Manton. The geometry of the Newton method on non-compact Lie groups. J. Global Optim. 23(3) (2002) 309-327.
R. Mahony. The constrained Newton method on a Lie group and the symmetric eigenvalue problem. Linear Algebra Appl. 248 (1996) 67-89.
J. H. Manton. Optimization algorithms exploiting unitary constraints. IEEE Trans. Signal Process. 50(3) (2002) 635-650.
B. Owren and B. Welfert. The Newton iteration on Lie groups. BIT 40(1) (2000) 121-145.
H. R. Rutishauser. Simultaneous iteration method for symmetric matrices. Numerische Mathematik 16 (1970) 205-223.
M. Shub. Some remarks on dynamical systems and numerical analysis. In VII ELAM. (1986) 69-92.
S. T. Smith. Geometric optimization methods for adaptive filtering. Ph.D. thesis (1993) Division of Applied Sciences, Harvard University.
S. T. Smith. Optimization techniques on Riemannian manifolds. Hamiltonian and gradient flows, algorithms and control, Fields Inst. Commun. 3. (1994) 113-136.
Y. Yang Optimization on Riemannian manifold. Proceedings of the 38th Conference on Decision and Control (1999)
Y. Yang Globally convergent optimization algorithms on Riemannian manifolds: Uniform framework for unconstrained and constrained optimization. Journal of Optimization Theory and Applications 132(2) (2007) 245-265.
