Improved smoothing Newton methods for symmetric cone complementarity problems

There recently has been much interest in smoothing Newton method for solving nonlinear complementarity problems. We extend such method to symmetric cone complementarity problems (SCCP). In this paper, we first investigate a one-parametric class of smoothing functions in the context of symmetric cones, which contains the Fischer–Burmeister smoothing function and the CHKS smoothing function as special cases. Then we propose a smoothing Newton method for the SCCP based on the one-parametric class of smoothing functions. For the proposed method, besides the classical step length, we provide a new step length and the global convergence is obtained. Finally, preliminary numerical results are reported, which show the effectiveness of the two step lengthes in the algorithm and provide efficient domains of the parameter for the complementarity problems.

[1]  伊師 英之,et al.  書評 J. Faraut and A. Koranyi: Analysis on Symmetric Cones (Oxford Math. Monogr.) , 2006 .

[2]  R. Sznajder,et al.  Some P-properties for linear transformations on Euclidean Jordan algebras , 2004 .

[3]  Stephen P. Boyd,et al.  Applications of second-order cone programming , 1998 .

[4]  Naihua Xiu,et al.  A Regularized Smoothing Newton Method for Symmetric Cone Complementarity Problems , 2008, SIAM J. Optim..

[5]  Naihua Xiu,et al.  New smooth C-functions for symmetric cone complementarity problems , 2007, Optim. Lett..

[6]  Jein-Shan Chen,et al.  A one-parametric class of merit functions for the symmetric cone complementarity problem , 2009 .

[7]  F. Facchinei,et al.  Finite-Dimensional Variational Inequalities and Complementarity Problems , 2003 .

[8]  Panos M. Pardalos,et al.  Nonlinear Analysis and Variational Problems , 2010 .

[9]  Panos M. Pardalos,et al.  Topics in Semidefinite and Interior-Point Methods , 1998 .

[10]  A new class of smoothing complementarity functions over symmetric cones , 2010 .

[11]  吉瀬 章子.  Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones , 2011 .

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

[13]  M. Seetharama Gowda,et al.  Some P-Properties for Nonlinear Transformations on Euclidean Jordan Algebras , 2005, Math. Oper. Res..

[14]  Deyun Wei,et al.  A new class of complementarity functions for symmetric cone complementarity problems , 2011, Optim. Lett..

[15]  J. Faraut,et al.  Analysis on Symmetric Cones , 1995 .

[16]  M. Seetharama Gowda,et al.  Some inertia theorems in Euclidean Jordan algebras , 2009 .

[17]  Takashi Tsuchiya,et al.  Primal-dual algorithms and infinite-dimensional Jordan algebras of finite rank , 2003, Math. Program..

[18]  Helmut Kleinmichel,et al.  A New Class of Semismooth Newton-Type Methods for Nonlinear Complementarity Problems , 1998, Comput. Optim. Appl..

[19]  Donald Goldfarb,et al.  Second-order cone programming , 2003, Math. Program..

[20]  P. Pardalos,et al.  Novel Approaches to Hard Discrete Optimization , 2003 .

[21]  Farid Alizadeh,et al.  Associative and Jordan Algebras, and Polynomial Time Interior-Point Algorithms for Symmetric Cones , 2001, Math. Oper. Res..

[22]  Farid Alizadeh,et al.  Extension of primal-dual interior point algorithms to symmetric cones , 2003, Math. Program..

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

[24]  L. Faybusovich,et al.  Jordan-Algebraic Aspects of Nonconvex Optimization over Symmetric Cones , 2006 .

[25]  Sangho Kum,et al.  Penalized complementarity functions on symmetric cones , 2010, J. Glob. Optim..

[26]  L. Faybusovich.  Euclidean Jordan Algebras and Interior-point Algorithms , 1997 .

[27]  Stephen P. Boyd,et al.  A primal—dual potential reduction method for problems involving matrix inequalities , 1995, Math. Program..