TY - JOUR
T1 - Differentiability v.s. convexity for complementarity functions
AU - Huang, Chien Hao
AU - Chen, Jein Shan
AU - Martinez-Legaz, Juan Enrique
N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.
PY - 2017/1/1
Y1 - 2017/1/1
N2 - 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.
AB - 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.
KW - Closed convex cone
KW - Complementarity functions
KW - NCP-functions
KW - Second-order cone
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U2 - 10.1007/s11590-015-0993-1
DO - 10.1007/s11590-015-0993-1
M3 - Article
AN - SCOPUS:84952654148
SN - 1862-4472
VL - 11
SP - 209
EP - 216
JO - Optimization Letters
JF - Optimization Letters
IS - 1
ER -