Differentiability v.s. convexity for complementarity functions

The J.-S. Chen's work is supported by Ministry of Science and Technology, Taiwan. The work of J. E. Martinez-Legaz has been supported by the MINECO of Spain, Grant MTM2014-59179-C2-2-P, and under Australian Research Council's Discovery Projects funding scheme (project number DP140103213). He is affiliated to MOVE (Markets, Organizations and Votes in Economics).It is known that complementarity functions play an important role in dealing with complementarity problems. The most well known complementarity problem is the symmetric cone complementarity problem (SCCP) which includes nonlinear complementarity problem (NCP), semidefinite complementarity problem (SDCP), and second-order cone complementarity problem (SOCCP) as special cases. Moreover, there is also so-called generalized complementarity problem (GCP) in infinite dimensional space. Among the existing NCP-functions, it was observed that there are no differentiable and convex NCP-functions. In particular, Miri and Effati (J Optim Theory Appl 164:723-730, 2015) show that convexity and differentiability cannot hold simultaneously for an NCP-function. In this paper, we further establish that such result also holds for general complementarity functions associated with the GCP