  The nonsmooth composite matrix optimization problem (CMatOP), in particular,
the matrix norm minimization problem, is a generalization of the matrix conic
programming problem with wide applications in numerical linear algebra,
computational statistics and engineering. This paper is devoted to the
characterization of the strong regularity for the CMatOP via the generalized
strong second order sufficient condition and constraint nondegeneracy for
problems with nonsmooth objective functions. The derived result supplements the
existing characterization of the strong regularity for the constrained
optimization problems with twice continuously differentiable data.
