Extended Lorentz cones and mixed complementarity problems

TL;DR

This paper extends Lorentz cones to define isotone projection sets, particularly Cartesian products, and applies this to iteratively solve general mixed complementarity problems.

Contribution

It introduces the concept of extended Lorentz cones and characterizes isotone projection sets, enabling new iterative solution methods for mixed complementarity problems.

Findings

Characterization of isotone projection sets with respect to extended Lorentz cones

Identification of Cartesian products as isotone projection sets

Development of an iterative approach for solving mixed complementarity problems

Abstract

In this paper we extend the notion of a Lorentz cone. We call a closed convex set isotone projection set with respect to a pointed closed convex cone if the projection onto the set is isotone (i.e., monotone) with respect to the order defined by the cone. We determine the isotone projection sets with respect to an extended Lorentz cone. In particular a Cartesian product between an Euclidean space and any closed convex set in another Euclidean space is such a set. We use this property to find solutions of general mixed complementarity problems in an iterative way.