X-ray absorption spectroscopy (XAS) is a powerful too to elucidate structural and dynamical information based on the localized core orbitals of atoms, molecules, solids, and materials. XAS provides element-specific information while maintaining sensitivity to chemical environment, and theoretical calculations of core-to-valence transition energies are invaluable for interpreting such spectra. Available computational models include time-dependent density functional theory (TD-DFT), orbital-optimized excited-state DFT, various correlated wave function models, or the Bethe-Salpeter equation (BSE) approach. Each of these methods is widely used and available in standard electronic structure codes, but there are limitations. Not least among these is cost, as only the DFT-based approaches are scalable to large systems such as proteins or liquid environments. Excitedstate DFT methods, which are based on finding a non-Aufbau solution to the Kohn-Sham equations, require a tedious state-by-state approach if an entire excitation spectrum is desired. TD-DFT can furnish the entire core-level spectrum in a single shot (when used with frozen occupied orbitals for the valence electrons), yet results tend to be much more functional-dependent even as compared to ground-state DFT. Building on previous work, we seek simplified approaches based on Kohn-Sham eigenvalues only, which encode information regarding chemical shifts and may be useful in modeling emerging transient spectroscopies at x-ray and extreme ultraviolet wavelengths. The present work investigates "core-hole constraining" methods for XAS, using time-independent (ground-state) DFT calculations. There are several variants, as described below, but the unifying feature of these methods is that an electron is promoted from a core orbital into a valence virtual orbital and then orbital relaxation is incorporated by solving the Kohn-Sham equations in the presence of a core hole, with the excitation energy ω i→f estimated as the difference between virtual and occupied energy levels,