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Characterizations of Tilt-Stable Local Minimizers of a Class of Matrix Optimization Problems

Chao Ding, Ebrahim Sarabi, Shiwei Wang

Abstract

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.

Characterizations of Tilt-Stable Local Minimizers of a Class of Matrix Optimization Problems

Abstract

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.

Paper Structure

This paper contains 6 sections, 20 theorems, 161 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Characterizations of Tilt-Stable Minimizers
  4. Quadratic Bundle of Spectral Functions
  5. Tilt-Stability in Matrix Optimization Problems
  6. Conclusion

Key Result

Proposition 2.3

Let $f\,\colon{\bf X}\to\overline{{\bf R}}$ be prox-regular and subdifferentially continuous at $\bar{x}$ for $\bar{v}=0$. Then the following conditions are equivalent.

Figures (2)

  • Figure 1: Different partitions of eigenvalues
  • Figure 2: $\alpha^l\times \alpha^l$ block of $P^\top HP$

Theorems & Definitions (33)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3: tilt stability via the second-order growth condition
  • Definition 3.1
  • Proposition 3.2
  • Remark 3.3
  • Proposition 3.4: second-order characterization of tilt stability
  • Theorem 3.5: characterization of tilt-stability via second subderivative
  • Definition 3.6
  • Proposition 3.7
  • ...and 23 more