The penalized Fischer‐Burmeister SOC complementarity function

In this paper, we study the properties of the penalized Fischer‐Burmeister (FB) second‐order cone (SOC) complementarity function. We show that the function possesses similar desirable properties of the FB SOC complementarity function for local convergence; for example, with the function the second‐order cone complementarity 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.