The Jacobian Consistency of a One-Parametric Class of Smoothing Functions for SOCCP

Abstract

Second-order cone (SOC) complementarity functions and their smoothing functions have been much studied in the solution of second-order cone complementarity problems (SOCCP). In this paper, we study the directional derivative and B-subdifferential of the one-parametric class of SOC complementarity functions, propose its smoothing function, and derive the computable formula for the Jacobian of the smoothing function. Based on these results, we prove the Jacobian consistency of the one-parametric class of smoothing functions, which will play an important role for achieving the rapid convergence of smoothing methods. Moreover, we estimate the distance between the subgradient of the one-parametric class of the SOC complementarity functions and the gradient of its smoothing function, which will help to adjust a parameter appropriately in smoothing methods.