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This note is concerned with the nonlinear Ky Fan $k$-norm matrix conicoptimization problems, which include the nuclear norm regularized minimizationproblem as a special case. For this class of nonpolyhedral matrix conicoptimization problems, under the assumption that a stationary solutionsatisfies the second-order sufficient condition and the associated Lagrangemultiplier satisfies the strict Robinson's CQ, we show that two classes ofperturbed KKT systems are locally upper Lipschitz at the origin, which impliesa local error bound for the distance from any point in a neighborhood of thecorresponding KKT point to the whole set of KKT points.

Locally upper Lipschitz of the perturbed KKT system of Ky Fan $k$-norm matrix conic optimization problems