Linear Complementarity Problems Over A Special Class Of Extended Second Order Cones

The complementarity problem is a cross-cutting area of research,which has a wide range of applications in economics,finance and other fields.In recent years second order cone complementarity problem has been used in machine learning,game theory,neural networks and so on.Nemeth and Zhang extended the notion of a second order cone in R1+q and obtained a pair of mutually dual extended second order cones in Rp+q denoted by L(p,q)and M(p,q)in 2015.They are both convex polyhedral cones if and only if q=1.In this paper,we first study some geometric properties of such the special extended second order cones L(p,1),give a necessary condition and a sufficient condition for the existence of solutions of the linear com-plementarity problems over L(p,1)and then present some necessary conditions for the globally uniquely solvable property of the linear complementarity problems over L(p,1)and a sufficient and necessary condition for the globally uniquely solvable property.Finally,we give an example which proves the identity transformation defined in R2+1 has the globally uniquely solvable property.