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

TL;DR

This paper establishes conditions for tilt stability in composite optimization problems involving the Ky-Fan κ-norm, providing practical criteria for identifying stable solutions in nonconvex, nonsmooth settings.

Contribution

It offers a new necessary and sufficient condition for tilt stability in Ky-Fan κ-norm composite problems, including practical criteria for nuclear-norm and spectral norm regularized minimizations.

Findings

Derived a verifiable criterion for tilt stability in Ky-Fan κ-norm problems.

Provided practical tests for stability in nuclear-norm regularized problems.

Extended tilt stability analysis to nonconvex, nonsmooth optimization contexts.

Abstract

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.