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

This note is concerned with the nonlinear Ky Fan $k$-norm matrix conic optimization problems, which include the nuclear norm regularized minimization problem as a special case. For this class of nonpolyhedral matrix conic optimization problems, under the assumption that a stationary solution satisfies the second-order sufficient condition and the associated Lagrange multiplier satisfies the strict Robinson’s CQ, we show that two classes of perturbed KKT systems are locally upper Lipschitz at the origin, which implies a local error bound for the distance from any point in a neighborhood of the corresponding KKT point to the whole set of KKT points.