A duality between the metric projection onto a convex cone and the metric projection onto its dual in Hilbert spaces

TL;DR

This paper establishes a duality relationship between metric projections onto convex cones and their duals in Hilbert spaces, extending finite-dimensional results using lattice operations.

Contribution

It generalizes a known Euclidean space duality result to infinite-dimensional Hilbert spaces through a novel proof approach.

Findings

Equivalence of isotonicity of P_K and subadditivity of P_L

Extension of Euclidean space results to Hilbert spaces

Use of extended lattice operations in the proof

Abstract

If $K$ and $L$ are mutually dual closed convex cones in a Hilbert space with the metric projections onto them denoted by $P_K$ and $P_L$ respectively, then the following two assertions are equivalent: (i) $P_K$ is isotone with respect to the order induced by $K$ (i. e. $v-u\in K$ implies $P_Kv-P_Ku\in K$); (ii) $P_L$ is subadditive with respect to the order induced by $L$ (i. e. $P_Lu+P_Lv-P_L(u+v)\in L$ for any $u, v \in \R^n$). This extends the similar result of A. B. N\'emeth and the author for Euclidean spaces. The extension is essential because the proof of the result for Euclidean spaces is essentially finite dimensional and seemingly cannot be extended for Hilbert spaces. The proof of the result for Hilbert spaces is based on a completely different idea which uses extended lattice operations.