Smoothing Functions for Second-Order-Cone Complementarity Problems

Smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints. In this paper, we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second-order cones. These smoothing functions include the Chen--Mangasarian class and the smoothed Fischer--Burmeister function. We study the Lipschitzian and differential properties of these functions and, in particular, we derive computable formulas for these functions and their Jacobians. These properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems. In particular, we establish the existence and uniqueness of the Newton direction when the underlying mapping is monotone.