Extended Lorentz cones and variational inequalities on cylinders

Positive operators of Extended Lorentz cones

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

Lorentz正锥在非凸二次优化中的应用

Applications of the Lorentz positive cone in nonconvex quadratic optimization

本文研究Lorentz正锥及其对偶锥，用于求解椭球约束下的双线性目标最小化问题，提供多项式时间算法，并增强非凸二次优化问题的SDP松弛。

A representation theorem for the Lorentz cone automorphisms

In this note we prove a representation theorem for the symmetric cone automorphisms in the spin algebra\, $\Ln$.

Extended Lorentz cones and mixed complementarity problems

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.

The dual Z-property for the Lorentz cone

The Z-property of a linear map with respect to a cone is an extension of the notion of Z-matrices. In a recent paper of Orlitzky (see Corollary 6.2 in M. Orlitzky. Positive and $\mathbf{Z}$-operators on closed convex cones, Electron. J Linear Algebra, 444--458, 2018) the characterisation of cone-complementarity is given in terms of the dual of the cone of linear maps satisfying the Z-property. Therefore, it is meaningful to consider the problem of finding the dual cone of the cone of linear maps which have the Z-property with respect to a cone. This short note will solve this problem in the pa…

The monotone extended second order cone and mixed complementarity problems

In this paper, we study a new generalization of the Lorentz cone, called the Monotone Extended Second Order Cone (MESOC). We investigate basic properties of MESOC including computation of its Lyapunov rank and proving its reducibility. Moreover, we show that in an ambient space, a cylinder is an isotonic projection set with respect to MESOC. We also examine a Nonlinear Complementarity Problem on a cylinder, which is equivalent to a suitable Mixed Complementarity Problem and provide a computational example illustrating applicability of MESOC.