  This article is concerned with the numerical solution of subspace
optimization problems, consisting of minimizing a smooth functional over the
set of orthogonal projectors of fixed rank. Such problems are encountered in
particular in electronic structure calculation (Hartree-Fock and Kohn-Sham
Density Functional Theory -DFT- models). We compare from a numerical analysis
perspective two simple representatives, the damped self-consistent field (SCF)
iterations and the gradient descent algorithm, of the two classes of methods
competing in the field: SCF and direct minimization methods. We derive
asymptotic rates of convergence for these algorithms and analyze their
dependence on the spectral gap and other properties of the problem. Our
theoretical results are complemented by numerical simulations on a variety of
examples, from toy models with tunable parameters to realistic Kohn-Sham
computations. We also provide an example of chaotic behavior of the simple SCF
iterations for a nonquadratic functional.
