In this paper, we conduct a thorough study on the first and second order properties of the Moreau-Yosida regularization of the vector $k$-norm function, the indicator function of its epigraph, and the indicator function of the vector $k$-norm ball. We start with settling the vector $k$-norm case via applying the existing breakpoint searching algorithms to the metric projector over its dual norm ball. In order to solve the other two cases, we propose algorithms of low computational cost for the metric projectors over four basic polyhedral convex cones. These algorithms are then used to compute the metric projector over the epigraph of the vector $k$-norm function (or the vector $k$-norm ball) and its directional derivative. Moreover, we completely characterize the differentiability of the proximal point mappings of the three vector $k$-norm related functions. The work done in this paper serves as a key step to understand the Moreau-Yosida regularization of the matrix Ky Fan $k$-norm related functions and thus provides us with fundamental tools to use the proximal point algorithms to solve large scale matrix optimization problems involving the matrix Ky Fan $k$-norm function.
Technical report, Department of Mathematics, National University of Singapore, March/2011