A unified variational framework for the inverse Kohn-Sham problem

The inverse Kohn-Sham (KS) problem seeks a local effective potential whose noninteracting ground state reproduces a prescribed electron density. Although many inversion formulations and schemes have been developed, they are often formulated in disparate languages, including reduced variational optimization, penalty regularization, response-based iteration, and PDE-constrained optimization. In this work, we develop a unified framework for inverse KS theory in two steps. First, we identify the fixed-density noninteracting constrained search and its density-potential duality as the natural variational anchor of the inverse KS problem. In this setting, the KS potential appears as the variational dual object associated with density reproduction, reducing to the familiar multiplier picture in regular regimes. Second, building on this anchor, we classify major inversion formulations according to how the KS state equations and density-reproduction condition are treated within the optimization architecture, with orbital orthonormality retained as an additional structural constraint. Within this framework, the Wu-Yang formulation appears as a potential-space reduced multiplier formulation, the Zhao-Morrison-Parr construction as a quadratic-penalty relaxation, and PDE-constrained approaches as explicit state-constraint formulations at the orbital level. Rather than comparing inversion formulations primarily at the level of implemented algorithms, the present work develops an optimization-theoretic formulation map. This viewpoint identifies where additive-constant ambiguity, asymptotic normalization, nonsmooth variational structure, metric choice, and weak-gap instability enter different inversion architectures, and it makes explicit how major inversion approaches are connected and where algorithmic design choices arise.