Abstract
The necessary condition established in Part I of this paper for the global maximizers of the maximization problem
over the Stiefel manifold \(\{ V \in \mathbb{R}^{m \times \ell } |V^ \top V = I_\ell \} (\ell < m)\), naturally leads to a self-consistent-field (SCF) iteration for computing a maximizer. In this part, we analyze the global and local convergence of the SCF iteration, and show that the necessary condition for the global maximizers is fulfilled at any convergent point of the sequences of approximations generated by the SCF iteration. This is one of the advantages of the SCF iteration over optimization-based methods. Preliminary numerical tests are reported and show that the SCF iteration is very efficient by comparing with some manifold-based optimization methods.
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Zhang, L., Li, R. Maximization of the sum of the trace ratio on the Stiefel manifold, II: Computation. Sci. China Math. 58, 1549–1566 (2015). https://doi.org/10.1007/s11425-014-4825-z
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DOI: https://doi.org/10.1007/s11425-014-4825-z
Keywords
- trace ratio
- Rayleigh quotient
- Stiefel manifold
- nonlinear eigenvalue problem
- optimality condition
- self-consistent-field iteration
- eigenspace