Properties of the solution set of generalized polynomial complementarity problems

Abstract

In this paper, we consider the {\it generalized polynomial complementarity problem} (GPCP), which covers the recently introduced {\it polynomial complementarity problem} (PCP) and the well studied {\it tensor complementarity problem} (TCP) as special cases. By exploiting the structure of tensors, we first show that the solution set of GPCPs is nonempty and compact when a pair of leading tensors is cone {\bf ER}. Then, we study some topological properties of the solution set of GPCPs under the condition that the leading tensor pair is cone ${\bf R}_0$. Finally, we study a notable global Lipschitzian error bound of the solution set of GPCPs, which is better than the results obtained in the current PCPs and TCPs literature. Moreover, such an error bound is potentially helpful for finding and analyzing numerical solutions to the problem under consideration.