Department of Applied Mathematics & Physics, Kyoto University

Technical Report 2000-009 (November 06, 2000)

Smoothing Functions for Second-Order-Cone Complementarity Problems
by Masao Fukushima, Zhi-Quan Luo, and Paul Tseng

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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 where 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 non-interior continuation methods for solving the corresponding optimization and complementarity problems. In particular, we establish existence and uniqueness of the Newton direction when the underlying mapping is monotone.