Existence and Uniqueness of Solutions for Homogeneous Cone Complementarity Problems

We consider existence and uniqueness properties of a solution to homogeneous cone complementarity problem. Employing an algebraic characterization of homogeneous cones due to Vinberg from the 1960s, we generalize the properties of existence and uniqueness of solutions for a nonlinear function associated with the standard nonlinear complementarity problem to the setting of homogeneous cone complementarity problem. We provide sufficient conditions for a continuous function so that the associated homogeneous cone complementarity problems have solutions. In particular, we give sufficient conditions for a monotone continuous function so that the associated homogeneous cone complementarity problem has a unique solution (if any). Moreover, we establish a global error bound for the homogeneous cone complementarity problem under some conditions.