A semismooth Newton method for SOCCPs based on a one-parametric class of SOC complementarity functions

摘要

In this paper, we present a detailed investigation for the properties of a one-parametric class of SOC complementarity functions, which include the globally Lipschitz continuity, strong semismoothness, and the characterization of their B-subdifferential. Moreover, for the merit functions induced by them for the second-order cone complementarity problem (SOCCP), we provide a condition for each stationary point to be a solution of the SOCCP and establish the boundedness of their level sets, by exploiting Cartesian P-properties. We also propose a semismooth Newton type method based on the reformulation of the nonsmooth system of equations involving the class of SOC complementarity functions. The global and superlinear convergence results are obtained, and among others, the superlinear convergence is established under strict complementarity. Preliminary numerical results are reported for DIMACS second-order cone programs, which confirm the favorable theoretical properties of the method. Keywords Second-order cone - Complementarity - B-subdifferential - Semismooth - Newton’s method S. Pan work is partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province. J.-S. Chen member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. The author’s work is partially supported by National Science Council of Taiwan.