Novel constructions of complementarity functions associated with symmetric cones

We provide affirmative answers to two long-standing questions regarding symmetric cone complementarity problem: (i) Is there systematic way to construct complementarity functions associated with symmetric cone? (ii) Is it possible to utilize existing NCP-functions to construct complementarity functions for symmetric cone? More specifically, we present three different assumptions, under one of which, we can construct complementarity functions associated with symmetric cone. For the second question, we demonstrate how to write out complementarity functions associated with symmetric cone by using a given NCP-function. Especially, we construct simple complementarity functions in the settings of second-order cone and positive semidefinite cone, which are two special types of symmetric cones. This novel idea opens up a new approach in solving the complementarity problem based on NCP-functions.