Complementarity Functions and Numerical Experiments on Some Smoothing Newton Methods for Second-Order-Cone Complementarity Problems

Two results on the second-order-cone complementarity problem are presented. We show that the squared smoothing function is strongly semismooth. Under monotonicity and strict feasibility we provide a new proof, based on a penalized natural complementarity function, for the solution set of the second-order-cone complementarity problem being bounded. Numerical results of squared smoothing Newton algorithms are reported.

[1]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[2]  Kim-Chuan Toh,et al.  SDPT3 -- A Matlab Software Package for Semidefinite Programming , 1996 .

[3]  Masao Fukushima,et al.  Smoothing Functions for Second-Order-Cone Complementarity Problems , 2002, SIAM J. Optim..

[4]  Defeng Sun,et al.  Sub-quadratic convergence of a smoothing Newton algorithm for the P0– and monotone LCP , 2004, Math. Program..

[5]  P. Pardalos,et al.  Handbook of global optimization , 1995 .

[6]  G. Isac.  Complementarity Problems , 1992 .

[7]  Defeng Sun,et al.  A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities , 2000, Math. Program..

[8]  Jean-Pierre Crouzeix,et al.  Pseudomonotone variational inequality problems: Existence of solutions , 1997, Math. Program..

[9]  L. Qi,et al.  Solving variational inequality problems via smoothing-nonsmooth reformulations , 2001 .

[10]  Xiaojun Chen,et al.  A penalized Fischer-Burmeister NCP-function , 2000, Math. Program..

[11]  K. G. Murty,et al.  Complementarity problems , 2000 .

[12]  Christian Kanzow,et al.  Semidefinite Programs: New Search Directions, Smoothing-Type Methods, and Numerical Results , 2002, SIAM J. Optim..

[13]  Liqun Qi,et al.  A nonsmooth version of Newton's method , 1993, Math. Program..

[14]  L. McLinden,et al.  Stable monotone variational inequalities , 1990, Math. Program..

[15]  Christian Kanzow,et al.  Complementarity And Related Problems: A Survey , 1998 .

[16]  Defeng Sun,et al.  A Squared Smoothing Newton Method for Nonsmooth Matrix Equations and Its Applications in Semidefinite Optimization Problems , 2003, SIAM J. Optim..

[17]  L. Faybusovich.  Linear systems in Jordan algebras and primal-dual interior-point algorithms , 1997 .

[18]  Defeng Sun,et al.  Semismooth Matrix-Valued Functions , 2002, Math. Oper. Res..