  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.
