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

Nonsmooth Composite Matrix Optimization: Strong Regularity, Constraint Nondegeneracy and Beyond