  Inverse Kohn-Sham (iKS) problems are needed to fully understand the
one-to-one mapping between densities and potentials on which Density Functional
Theory is based. They are also important to advance computational schemes that
rely on density-to-potential inversions such as the Optimized Effective
Potential method and various techniques for density-based embedding. Unlike the
forward Kohn-Sham problems, numerical iKS problems are ill-posed and can be
unstable. We discuss some of the fundamental and practical difficulties of iKS
problems with constrained-optimization methods on finite basis sets. Various
factors that affect the performance are systematically compared and discussed,
both analytically and numerically, with a focus on two of the most practical
methods: the Wu-Yang method (WY) and partial-differential-equation
constrained-optimization (PDE-CO). Our analysis of the WY and PDE-CO highlights
the limitation of finite basis sets and the importance of regularization. We
introduce two new ideas that will hopefully contribute to making iKS problems
more tractable: (1) A correction to the WY method that utilizes the null space
of the relevant Hessian matrices; and (2) A finite potential basis-set
implementation of the PDE-CO method. We provide an overall strategy for
performing numerical density-to-potential inversions that can be directly
adopted in practice. We also provide an Appendix with several examples that can
be used for benchmarking.
