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

This paper studies the linear complementarity problem LCP(M, q) over the second-order (Lorentz or ice-cream) cone denoted by Λ+n, where M is a n × n real square matrix and q ∈ 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 Λ+n may yield interesting results. In this paper we characterize the R0-property (x∈Λ+n, M(x)∈Λ+n and 〈x, M(x)〉 = 0 ⇒ x = 0) of a quadratic representation Pa(x) := 2a ∘ (a ∘ x) − a2 ∘ x of Λn for a, x ∈ Λn where '∘' 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 ∈ Λn.