On the Coderivative of the Projection Operator onto the Positive Cone in Hilbert spaces

TL;DR

This paper derives explicit formulas for the generalized derivatives of the projection operator onto the positive cone in Hilbert spaces, enhancing understanding of its sensitivity and stability properties in infinite-dimensional settings.

Contribution

It provides the first explicit formulas for the regular and Mordukhovich coderivatives of the projection onto the positive cone in Hilbert spaces, extending Euclidean results.

Findings

Formulas for the regular coderivative in Euclidean spaces

Formulas for the Mordukhovich coderivative in Euclidean spaces

Extension of these formulas to the Hilbert space l_2

Abstract

In this paper, we study the generalized differentiability of the metric projection operator onto the positive cone in Hilbert spaces. We first establish the formula for exactly computing the regular coderivative and the Mordukhovich coderivative of the metric projection operator onto the positive cone in Euclidean spaces. Then, these results are also established for the projection operator onto the positive cone in the real Hilbert space $l_2$.