Extended Lorentz cones and 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.Comment: 15 page

How to project onto extended second order cones

“…Hence, λ j+1 = λ j+2 and in view of (20) we conclude that ψ(λ j+1 ) = 0 and λ j+1 is the solution of (5), i.e., λ j+1 = λ * . The next proposition shows that under a further restriction on the point which is projected the convergence of the semi-smooth Newton sequence is linear.…”

“…If we allow q = 0 as well, then the cones L and M reduce to the nonnegative orthant. More properties of the extended second order cones can be found in [19,20,25].…”

“…A nice property of the second order cone is that it admits an explicit representation of the projection mapping onto it (see [6,Proposition 3.3]). The original motivation for extending the second order cone was inspired by using iterative methods for solving complementarity problems and variational inequalities [20,21]. These iterative methods are based on the property that the projection onto the closed convex set defining the problem is isotone with respect to the order defined out by a cone.…”

“…However, cylinders and in particular cylinders with cone base admit isotone projections onto them with respect to the extended second order cones. Therefore, variational inequalities on cylinders and mixed complementarity problems can be solved by using such iterative techniques based on monotone convergence [19,20].…”

Linear Complementarity Problems on Extended Second Order Cones

“…Denote by · the corresponding Euclidean norm. Recall the definitions of the mutually dual extended second order cone [9] L(k, ) and M (k, ) in R n ≡ R k × R :…”