Positive operators on extended second order cones

Abstract

In this paper necessary conditions and sufficient conditions are given for a linear operator to be a positive operator of an Extended Lorentz cone. Similarities and differences with the positive operators of Lorentz cones are investigated. * 2010

How to project onto extended second order cones

“…By letting µ = λ * and λ = λ k in inequality (19) and by using again the definition in (17), we obtain that…”

“…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].…”

“…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].…”

The Monotone Extended Second-Order Cone and Mixed Complementarity Problems

“…≥+ and (R n ≥+ ) * , respectively, and x ⊥ y. As Au i , v j = 0, by considering the derivation of equations ( 9), (10), (11) and (12) in the reverse order, equation (13) implies that Ax, y = 0. Hence, A ∈ LL(R n ≥+ ).…”

“…There are several known important versions of the extended Lorentz cone, including Bishop-Phelps cone [5] and the extended second-order cone (ESOC), which was recently developed by S. Z. Németh and his co-authors, see [10][11][12][13][14]. The Lyapunov rank of a cone K , denoted by β(K ) (see its definition in the next section) is an invariant which shows that the Lorentz cone and ESOC are generally not linearly isomorphic.…”