The convex and monotone functions associated with second-order cone

Like the matrix-valued functions used in solutions methods for semidefinite programs (SDPs) and semidefinite complementarity problems (SDCPs), the vector-valued functions associated with second-order cones are defined analogously and also used in solutions methods for second-order-cone programs (SOCPs) and second-order-cone complementarity problems (SOCCPs). In this article, we study further about these vector-valued functions associated with second-order cones (SOCs). In particular, we define the so-called SOC-convex and SOC-monotone functions for any given function f: R -> R. We discuss the SOC-convexity and SOC-monotonicity for some simple functions, e.g., f(t) = t(2), t(3), 1/t, t(1/2). vertical bar t vertical bar, and [t](+). Some characterizations of SOC-convex and SOC-monotone functions are studied, and some conjectures about the relationship between SOC-convex and SOC-monotone functions are proposed.