Regular Coderivative and Graphical Derivative of the Metric Projection onto closed Balls in Hilbert spaces

TL;DR

This paper derives explicit formulas for the regular coderivative and graphical derivative of the metric projection operator onto closed balls in Hilbert spaces, advancing the theoretical understanding of these operators.

Contribution

It provides the first exact formulas for the regular coderivative and graphical derivative of metric projections onto closed balls in Hilbert spaces.

Findings

Explicit formula for regular coderivative onto centered balls

Extension of formulas to arbitrary centers in Hilbert spaces

Formula for graphical derivative at arbitrary points

Abstract

In this paper, we first establish a formula for exactly computing the regular coderivative of the metric projection operator onto closed balls $r\mathbb{B}$ centered at the origin in Hilbert spaces. Then, this result is extended to metric projection operator onto any closed balls $\mathbb{B}(c,r)$, which has center $c$ in Hilbert space $H$ and with radius $r > 0$. Finally, we give the formula for calculating the graphical derivative of the metric projection operator onto closed balls with center at arbitrarily given point in Hilbert spaces.