Tilt stability of Ky-Fan $κ$-norm composite optimization
Yulan Liu, Shaohua Pan, Wen Ping Song · arXiv (Cornell University) · 2024
This paper concerns the tilt stability for the minimization of the sum of a twice continuously differentiable matrix-valued function and the Ky-Fan $κ$-norm. To achieve this goal, we first provide a sufficient and necessary condition for a local minimizer of the composite $f=φ+g$ to be tilt-stable with the second subderivative of $g$, where $g$ is a closed proper convex function, and $φ$ 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 $κ$-norm composite problem, and employ the expression of second subderivative of the Ky-Fan $κ$-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.