Coercivity and Strong Semismoothness of the Penalized Fischer-Burmeister Function for the Symmetric Cone Complementarity Problem

For the nonlinear complementarity problem (NCP), Chen et al. (Math. Program., 88:211–216, 2000) proposed a penalized Fischer-Burmeister (FB) function that has most desirable properties among complementarity functions (C-functions). Motivated by their work, the authors showed (Kum and Lim in Penalized Complementarity Functions on Symmetric Cones, submitted, 2009) that this function naturally extends to a C-function for the symmetric cone complementarity problem (SCCP). In this note, we show that the main coercivity property of this function for NCP also extends to the SCCP. The proof uses a new trace inequality on Euclidean Jordan algebras. We also show that the penalized FB function is strongly semismooth in the case of a semidefinite cone and a second-order cone. This work was supported by the Korea Research Foundation Grant KRF-2008-314-C00039.