Comput Optim Appl (2011) 49:457–491 DOI 10.1007/s10589-009-9301-2 The penalized Fischer-Burmeister SOC complementarity function Shaohua Pan · Jein-Shan Chen · Sangho Kum · Yongdo Lim Received: 23 May 2008 / Published online: 20 November 2009 © Springer Science+Business Media, LLC 2009 Abstract In this paper, we study the properties of the penalized Fischer-Burmeister (FB) second-order cone (SOC) complementarity function. We show that the func- tion possesses similar desirable properties of the FB SOC complementarity function for local convergence; for example, with the function the second-order cone comple- mentarity problem (SOCCP) can be reformulated as a (strongly) semismooth system of equations, and the corresponding nonsmooth Newton method has local quadratic convergence without strict complementarity of solutions. In addition, the penalized FB merit function has bounded level sets under a rather weak condition which can be satisfied by strictly feasible monotone SOCCPs or SOCCPs with the Cartesian R 01 - property, although it is not continuously differentiable. Numerical results are included to illustrate the theoretical considerations. Work of S. Pan is supported by National Young Natural Science Foundation (No. 10901058) and Guangdong Natural Science Foundation (No. 9251802902000001). J.-S. Chen is a member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan. S. Pan School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China e-mail: shhpan@scut.edu.cn J.-S. Chen ( ) Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan e-mail: jschen@math.ntnu.edu.tw S. Kum Department of Mathematics Education, Chungbuk National University, Cheongju 361-763, Korea Y. Lim Department of Mathematics, Kyungpook National University, Taegu 702-701, Korea