ON THE MOREAU--YOSIDA REGULARIZATION OF THE VECTOR k-NORM RELATED FUNCTIONS.

Matrix optimization problems (MOPs) involving the Ky Fan k-norm arise frequently from many applications. In order to design algorithms to solve large scale MOPs involving the Ky Fan k-norm, we need to understand the first and second order properties of the Moreau-Yosida regularization of the Ky Fan k-norm function and the indicator function of the Ky Fan k-norm ball. According to the general theory on spectral functions, in this paper we shall conduct a thorough study on the Moreau--Yosida regularization of the vector k-norm function and the indicator function of the vector k-norm ball. In particular, we show that the proximal mappings associated with these two vector k-norm related functions both admit fast and analytically computable solutions. Moreover, we propose algorithms of low computational cost to compute the directional derivatives of these two proximal mappings and then completely characterize their Frechet differentiability. The work here thus builds the fundamental tools needed in the design of proximal point based algorithms for solving large scale MOPs involving the Ky Fan k-norm as well as in the study of the sensitivity and stability analysis of these problems. [ABSTRACT FROM AUTHOR]