The Wannier localization problem plays an important role in Hartree-Fock and Kohn-Sham density functional theory calculations. In particular, the ability to robustly localize orbitals is an important step in a wide range of accelerated computational schemes. This problem is often framed as a non-convex optimization problem, and in the condensed phase the natural objective function needs to be properly adapted for the boundary conditions. We will present a new perspective on how to derive several commonly used objective functions and highlight previously unknown relations between them. Additionally, we will leverage our insights to illustrate that the choice of objective function has a significant impact on the practical computation of localized functions. Lastly, we will briefly discuss the consequences of these developments when computing a localized basis for unoccupied orbitals, highlight advances in schemes to initialize the localization problem, and discuss recent software developments.
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