New formulas for subdifferentials of perturbed distance functions

ABSTRACT

We give exact formulas for the subdifferentials of perturbed distance functions in a normed space. Our method, seemingly novel and different from existing ones, is to turn the involved problem equivalently to a parametric optimization problem and then apply variational analysis technique to the optimal value function. In the convex setting, we obtain new representations for the subdifferential of perturbed distance functions, which do not depend on the relative position of the reference point with respect to the input set, and which are described directly via the input data. Our results complement those of Wang et al. [J. Global Optim. 2010;46:489–501] and of Li and Bounkhel [Nonlinear Anal. 2014;108:173–188], which were established by different methods.

KEYWORDS:

• Perturbed optimization problem
• perturbed distance function
• distance function
• optimal value function
• subdifferential

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 90C30
• 90C31
• 90C56
• 49J52
• 49J53
• 49K27

Acknowledgements

The authors would like to thank the two anonymous referees for their comments and suggestions, which improved the presentation of this manuscript.

Disclosure statement

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

Additional information

Funding

Hong-Kun Xu was supported in part by the National Natural Science Foundation of China [grant number U1811461] and by the Australian Research Council [grant number DP200100124].