Isotonicity of the projection onto the monotone cone

TL;DR

This paper characterizes the conditions under which the projection onto certain cones, like the monotone cone, preserves order relations, with implications for regression and optimization.

Contribution

It provides a characterization of wedges that admit isotone metric projections, including the monotone cone used in regression theory.

Findings

Monotone cone admits isotone projection.

Characterization of wedges with isotone projections.

Application to regression and order-preserving methods.

Abstract

A wedge (i.e., a closed nonempty set in the Euclidean space stable under addition and multiplication with non-negative scalars) induces by a standard way a semi-order (a reflexive and transitive binary relation) in the space. The wedges admitting isotone metric projection with respect to the semi-order induced by them are characterized. The obtained result is used to show that the monotone wedge (called monotone cone in regression theory) admits isotone projection.