Lattice-like operations and isotone projection sets

TL;DR

This paper introduces lattice-like operations to analyze the isotonicity of metric projections on convex sets, extending previous results for self-dual cones and exploring applications in variational inequalities.

Contribution

It generalizes existing results by defining new lattice-like operations and characterizing convex sets with isotonic metric projections, broadening the theoretical framework.

Findings

Generalized results for self-dual cones

Identified convex sets with isotonic projections

Linked isotonicity to variational inequalities

Abstract

By using some lattice-like operations which constitute extensions of ones introduced by M. S. Gowda, R. Sznajder and J. Tao for self-dual cones, a new perspective is gained on the subject of isotonicity of the metric projection onto the closed convex sets. The results of this paper are wide range generalizations of some results of the authors obtained for self-dual cones. The aim of the subsequent investigations is to put into evidence some closed convex sets for which the metric projection is isotonic with respect the order relation which give rise to the above mentioned lattice-like operations. The topic is related to variational inequalities where the isotonicity of the metric projection is an important technical tool. For Euclidean sublattices this approach was considered by G. Isac and respectively by H. Nishimura and E. A. Ok.