Mixed lattice structures and cone projections

TL;DR

This paper introduces a new order-theoretic framework for cone projection problems in optimization, generalizing mixed lattice spaces with two partial orderings to unify various approaches.

Contribution

It proposes a novel perspective using a generalized mixed lattice space structure based on two partial orderings, broadening the mathematical tools for cone projection problems.

Findings

Generalized mixed lattice spaces naturally arise in cone projection applications.

The order-theoretic formalism unifies existing approaches in optimization.

New lattice-like properties are identified in the generalized structures.

Abstract

Problems related to projections on closed convex cones are frequently encountered in optimization theory and related fields. To study these problems, various unifying ideas have been introduced, including asymmetric vector-valued norms and certain generalized lattice-like operations. We propose a new perspective on these studies by describing how the problem of cone projection can be formulated using an order-theoretic formalism developed in this paper. The underlying mathematical structure is a partially ordered vector space that generalizes the notion of a vector lattice by using two partial orderings and having certain lattice-type properties with respect to these orderings. In this note we introduce a generalization of these so-called mixed lattice spaces, and we show how such structures arise quite naturally in some of the applications mentioned above.