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Moreau--Yosida regularization in DFT

Moreau-Yosida regularization is introduced into the framework of exact DFT. Moreau-Yosida regularization is a lossless operation on lower semicontinuous proper convex functions over separable Hilbert spaces, and when applied to the universal functional of exact DFT (appropriately restricted to a bounded domain), gives a reformulation of the ubiquitous $v$-representability problem and a rigorous and illuminating derivation of Kohn-Sham theory. The chapter comprises a self-contained introduction to exact DFT, basic tools from convex analysis such as sub- and superdifferentiability and convex conjugation, as well as basic results on the Moreau-Yosida regularization. The regularization is then applied to exact DFT and Kohn-Sham theory, and a basic iteration scheme based in the Optimal Damping Algorithm is analyzed. In particular, its global convergence established. Some perspectives are offered near the end of the chapter.