Tilt stability of Ky-Fan $\kappa$-norm composite optimization

This paper concerns the tilt stability for the minimization of the sum of a twice continuously differentiable matrix-valued function and the Ky-Fan $\kappa$-norm. To achieve this goal, we first provide a sufficient and necessary condition for a local minimizer of the composite $f=\varphi+g$ to be tilt-stable with the second subderivative of $g$, where $g$ is a closed proper convex function, and $\varphi$ is a twice continuously differentiable function that is locally convex at the local minimizer. Then, we apply the sufficient and necessary condition to the concerned Ky-Fan $\kappa$-norm composite problem, and employ the expression of second subderivative of the Ky-Fan $\kappa$-norm to derive a verifiable criterion to identify the tilt stability of a local minimum for this class of nonconvex and nonsmooth problems. As a byproduct, a practical criterion is obtained for identifying the tilt stablity of solutions to the nuclear-norm and spectral norm regularized minimization problems.