On Q and R0 properties of a quadratic representation in linear complementarity problems over the second-order cone Original Research Article

This paper studies the linear complementarity problem LCP(M, q) over the second-order (Lorentz or ice-cream) cone denoted by image, where M is a n × n real square matrix and q set membership, variant Rn. This problem is denoted as SOLCP(M, q). The study of second-order cone programming problems and hence an independent study of SOLCP is motivated by a number of applications. Though the second-order cone is a special case of the cone of squares (symmetric cone) in a Euclidean Jordan algebra, the geometry of its faces is much simpler and hence an independent study of LCP over image may yield interesting results. In this paper we characterize the R0-property (image, image and left angle bracketx, M(x)right-pointing angle bracket = 0 implies x = 0) of a quadratic representation Pa(x) := 2a ring operator (a ring operator x) − a2 ring operator x of Λn for a, x set membership, variant Λn where ‘ring operator’ is a Jordan product and show that the R0-property of Pa is equivalent to stating that SOLCP(Pa, q) has a solution for all q set membership, variant Λn.