Characterizations of Tilt-Stable Local Minimizers of a Class of Matrix Optimization Problems
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院数学科学重点实验室）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院应用数学研究所）；Department of Mathematics, Miami University（迈阿密大学数学系）；Institute of Operational Research and Analytics, National University of Singapore（新加坡国立大学运筹分析与研究院）
Tilt stability plays a pivotal role in understanding how local solutions of an optimization problem respond to small, targeted perturbations of the objective. Although quadratic bundles are a powerful tool for capturing second-order variational behavior, their characterization remains incomplete beyond well-known polyhedral and certain specialized nonpolyhedral settings. To help bridge this gap, we propose a new point-based criterion for tilt stability in prox-regular, subdifferentially continuous functions by exploiting the notion of minimal quadratic bundles. Furthermore, we derive an explicit formula for the minimal quadratic bundle associated with a broad class of general spectral functions, thus providing a practical and unifying framework that significantly extends existing results and offers broader applicability in matrix optimization problems.
