Fully stable well-posedness and fully stable minimum with respect to an admissible function

ABSTRACT

In this paper, the notions of fully stable well-posedness and fully stable minimum of optimization problems with an extended real-valued objective function in an Asplund space setting are considered, where the objective function undergoes both tilt perturbations and general parameter perturbations. Special cases of the first notion reduce to the stable Hölder minimizers by Zheng and Ng [SIAM J Optim. 2015;25:416–438] and the fully stable Hölder minimizers introduced by Zheng, Zhu and Ng [SIAM J Optim. 2018;28:2601–2624], respectively. By using techniques of variational analysis and generalized differentiation, the results presented in this paper provide insight into necessary or sufficient conditions for tilt-stable minimizers of nonlinear programmes in Asplund spaces and therefore generalize some existing ones in the recent literature.

KEYWORDS:

• Admissible function
• fully stable well-posedness
• fully stable minimum
• tilt stability

AMS CLASSIFICATIONS:

• 90C31
• 90C30
• 49K40
• 49J52

Acknowledgments

The authors are grateful to the referees and editors for their helpful comments and constructive suggestions which helped us to improve the presentation of the paper. Furthermore, the authors wish to express their sincere gratitude to Prof. Jen-Chih Yao for initiating this collaboration.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the National Natural Science Foundation of China [grant numbers 11801497 and 11771384].