Order isotonicity of the metric projection onto a closed convex cone

TL;DR

This paper investigates which proper cones in Euclidean space allow for metric projections that are order-preserving, extending previous work on coordinate-wise orderings to more general cone-induced orders.

Contribution

It characterizes the proper cones in Euclidean space that admit isotone metric projections with respect to the cone-induced order relation.

Findings

Identifies conditions under which metric projections are order-preserving.

Extends previous results from coordinate-wise to general cone-induced orders.

Provides a framework for solving isotonic regression problems in more general settings.

Abstract

The basic tool for solving problems in metric geometry and isotonic regression is the metric projection onto closed convex cones. Isotonicity of these projections with respect to a given order relation can facilitate finding the solutions of the above problems. In the recent note "A. B. N\'emeth and S.Z. N\'emeth: Isotonic regression and isotonic projection. Linear Algebra and its Applications, 494: 80-89 (2016)" this problem was studied for the coordinate-wise ordering. This study was the starting point for further investigations, such as the ones presented here. The order relation in the Euclidean space endowed by a proper cone is considered and the proper cones admitting isotone metric projections with respect to this order relation are investigated.