New formulas for subdifferentials of perturbed distance functions

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.

Automated summary

We provide exact formulas for the subdifferentials of perturbed distance functions in a normed space. Our novel method converts the problem into a parametric optimization problem and applies variational analysis technique to the optimal value function. In the convex setting, we obtain new representations for the subdifferential of perturbed distance functions that are independent of the relative position of the reference point and described directly by the input data. These results complement previous findings by Wang et al. and Li and Bounkhel, which were established using different methods.