Convergence Analysis of Direct Minimization and Self-Consistent Iterations

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