Solvability of implicit semidefinite and implicit copositive complementarity problems

In this paper, we introduce the concept of exceptional family to implicit semidefinite complementarity problems and implicit copositive complementarity problems. Based on the notion of the exceptional family of elements, we prove that the nonexistence of exceptional family of elements is a sufficient condition for the existence theorem of implicit semidefinite complementarity problems and the implicit copositive complementarity problems. The condition is also necessary for relatively pseudomonotone operators to implicit semidefinite complementarity problems. Our results generalize the corresponding results of Isac et al. (1997) and extend the results of Zhang (2008), Hu et al. (2012), Huang and Ma (2014) and Bulavsky et al. (2001). Moreover, we also present some theorems correlated to the structure and strict feasibility of implicit semidefinite complementarity problem. In our analysis, the new concept of exceptional family plays a vital role for the solvability of implicit semidefinite and implicit copositive complementarity problems. (C) 2020 Elsevier B.V. All rights reserved.

Automated summary

This paper introduces the concept of exceptional family to implicit semidefinite complementarity problems and copositive complementarity problems, proving that the nonexistence of exceptional family is a sufficient condition for the existence theorem. The new concept plays a vital role for the solvability of these problems and extends previous research in the field.