ZHANG Li-ping, GAO Zi-you. Global Linear and Quadratic One-Step Smoothing Newton Method for Vertical Linear Complementarity Problems[J]. Applied Mathematics and Mechanics, 2003, 24(6): 653-660.
 Citation: ZHANG Li-ping, GAO Zi-you. Global Linear and Quadratic One-Step Smoothing Newton Method for Vertical Linear Complementarity Problems[J]. Applied Mathematics and Mechanics, 2003, 24(6): 653-660.

# Global Linear and Quadratic One-Step Smoothing Newton Method for Vertical Linear Complementarity Problems

• Rev Recd Date: 2003-03-15
• Publish Date: 2003-06-15
• A one-step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so-called aggregation function. The proposed algorithm has the following good features: (i) it solves only one linear system of equations and does only one line search at each iteration; (ⅱ) it is well-defined for the vertical linear complementarity problem with vertical block P0 matrix and any accumulation point of iteration sequence is its solution. Moreover, the iteration sequence is bounded for the vertical linear complementarity problem with vertical block P0+R0 matrix; (ⅲ) it has both global linear and local quadratic convergence without strict complementarity. Many existing smoothing Newton methods do not have the property(ⅲ).
•  [1] Cottle R W,Pang J S,Stone R W.The Linear Complementarity Problem[M].Boston:Academic Press,1992. [2] Cottle R W,Dantzig G B.A generalization of the linear complementarity problem[J].Journal of Combinatorial Theory,1970,8(1):79-90. [3] Sun M.Monotonicity of mangsarian's iterative algorithm for generalized linear complementarity problems[J].J Math Anal Appl,1989,144(2):474-485. [4] Gowda M S,Sznajder R.A generalization of the Nash equilibrium theorem on bimatrix games[J].Internat J Game Theory,1996,25(1):1-12. [5] Ebiefung A A,Kostreva M M.The generalized Leontief input-output model and its application to the choice of the new technology[J].Ann Oper Res,1993,44(1):161-172. [6] Fujisawa T,Kuh E S.Piecewise-linear theory of nonlinear networks[J].SIAM J Appl Math,1972,22(1):307-328. [7] Peng J M,Lin Z H.A noninterior continuation method for generalized linear complementarity problems[J].Math Programming,1999,86(2):533-563. [8] Qi H,Liao L Z.A smoothing Newton method for extended vertical linear complementarity problems[J].SIAM J Matrix Anal Appl,1999,21(1):45-66. [9] Billups S C,Dirkse S P,Ferris M C.A Comparison of algorithms for large-scale mixed complementarity problems[J].Computational Optimization and Applications,1997,7(1):3-25. [10] Qi L,Sun D,Zhou G.A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems[J].Math Programming,2000,87(1):1-35. [11] Zhou G,Sun D,Qi L.Numerical experiences for a class of squared smoothing Newton methods for box constrained variational inequality problems[A].In:Fukushima M,Qi L Eds.Reformulation-Nonsmooth,Piecewise Smooth,Semismooth and Smoothing Methods[C].Boston:Kluwer Academic Publishers,1998,421-441. [12] Qi H.A regularized smoothing Newton method for box constrained variational inequality problems with P0-functions[J].SIAM Journal on Optimization,2000,10(1):315-330. [13] Sun D.A regularization Newton method for solving nonlinear complementarity problems[J].Applied Mathematics and Optimization,1999,35(1):315-339. [14] Chen X,Qi L,Sun D.Global and superlinear convergence of the smoothing Newton method and its application to general box-constrained variational inequalities[J].Mathematics of Computation,1998,67(2):519-540. [15] Kanzow C,Pieper H.Jacobian smoothing methods for nonlinear complementarity problems[J].SIAM Journal on Optimization,1999,9(1):342-373. [16] LI X S.An aggregate function method for nonlinear programming[J].Science in China, Ser A,1991,34(3):1467-1473. [17] Qi L,Sun J.A nonsmooth version of Newton's method[J].Math Programming,1993,58(1):353-367. [18] Qi L.Convergence analysis of some algorithms for solving nonsmooth equations[J].Mathematics of Operations Research,1993,18(1):227-244. [19] Chen B,Xiu N.Superlinear noninterior one-step continuation method for monotone LCP in absence of strict complementarity[J].Journal of Optimization Theory and Applications,2001,108(1):317-332.

### Catalog

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142