Smoothing Functions for Second-Order-Cone Complementarity Problems

TLDR

Smoothing functions have been extensively studied for solving optimization and complementarity problems with nonnegativity constraints. The paper extends smoothing functions to problems where the nonnegative orthant is replaced by a direct product of second‑order cones. The authors analyze the Chen–Mangasarian and smoothed Fischer–Burmeister smoothing functions, deriving Lipschitzian and differential properties, computable formulas for the functions and their Jacobians, and applying these results to design noninterior continuation methods for the resulting optimization and complementarity problems. They prove that, for monotone mappings, the Newton direction exists uniquely.

Abstract

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.