A Penalized Fischer-Burmeister Ncp-Function: Theoretical Investigation And Numerical Results

We introduce a new NCP-function that reformulates a nonlinear com-plementarity problem as a system of semismooth equations (x) = 0. The new NCP-function possesses all the nice properties of the Fischer-Burmeister function for local convergence. In addition, its natural merit function (x) = 1 2 (x) T (x) has all the nice features of the Kanzow-Yamashita-Fukushima merit function for global convergence. In particular, the merit function has bounded level sets for a monotone complementarity problem with a strictly feasible point. This property allows the existing semismooth Newton methods to solve this important class of comple-mentarity problems without additional assumptions. We investigate the properties of a semismooth Newton-type method based on the new NCP-function and apply the method to a large class of complementarity problems. The numerical results indicate that the new algorithm is extremely promising.

[1]  R. Mifflin.  Semismooth and Semiconvex Functions in Constrained Optimization , 1977 .

[2]  Stephen M. Robinson,et al.  Strongly Regular Generalized Equations , 1980, Math. Oper. Res..

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

[4]  L. Grippo,et al.  A nonmonotone line search technique for Newton's method , 1986 .

[5]  Shinji Mizuno,et al.  Limiting Behavior of Trajectories Generated by a Continuation Method for Monotone Complementarity Problems , 1990, Math. Oper. Res..

[6]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[7]  A. Fischer.  A special newton-type optimization method , 1992 .

[8]  Liqun Qi,et al.  Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations , 1993, Math. Oper. Res..

[9]  S. Dirkse,et al.  Mcplib: a collection of nonlinear mixed complementarity problems , 1995 .

[10]  S. Dirkse,et al.  Solver: A Non-Monotone Stabilization Scheme for Mixed Complementarity Problems , 1995 .

[11]  S. Billups.  Algorithms for complementarity problems and generalized equations , 1996 .

[12]  Andreas Fischer,et al.  Solution of monotone complementarity problems with locally Lipschitzian functions , 1997, Math. Program..

[13]  Francisco Facchinei,et al.  A semismooth equation approach to the solution of nonlinear complementarity problems , 1996, Math. Program..

[14]  P. Tseng.  Growth behavior of a class of merit functions for the nonlinear complementarity problem , 1996 .

[15]  Olvi L. Mangasarian,et al.  A class of smoothing functions for nonlinear and mixed complementarity problems , 1996, Comput. Optim. Appl..

[16]  M. Fukushima.  Merit Functions for Variational Inequality and Complementarity Problems , 1996 .

[17]  Christian Kanzow,et al.  Some Noninterior Continuation Methods for Linear Complementarity Problems , 1996, SIAM J. Matrix Anal. Appl..

[18]  Michael C. Ferris,et al.  ACCESSING REALISTIC MIXED COMPLEMENTARITY PROBLEMS WITHIN MATLAB , 1996 .

[19]  Patrick T. Harker,et al.  Smooth Approximations to Nonlinear Complementarity Problems , 1997, SIAM J. Optim..

[20]  Francisco Facchinei,et al.  A New Merit Function For Nonlinear Complementarity Problems And A Related Algorithm , 1997, SIAM J. Optim..

[21]  Xiaojun ChenyMay.  A Global and Local Superlinear Continuation-Smoothing Method for P0 +R0 and Monotone NCP , 1997 .

[22]  Christian Kanzow,et al.  A new approach to continuation methods for complementarity problems with uniform P-functions , 1997, Oper. Res. Lett..

[23]  M. Fukushima,et al.  New NCP-Functions and Their Properties , 1997 .

[24]  Houyuan Jiang,et al.  Semismooth Karush-Kuhn-Tucker Equations and Convergence Analysis of Newton and Quasi-Newton Methods for Solving these Equations , 1997, Math. Oper. Res..

[25]  F. Facchinei.  Structural and Stability Properties of P 0 Nonlinear Complementarity Problems , 1998 .

[26]  Houyuan Jiang,et al.  Global and Local Superlinear Convergence Analysis of Newton-Type Methods for Semismooth Equations with Smooth Least Squares , 1998 .

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

[28]  Defeng Sun,et al.  A New Unconstrained Differentiable Merit Function for Box Constrained Variational Inequality Problems and a Damped Gauss-Newton Method , 1999, SIAM J. Optim..

[29]  Y. Ye,et al.  On Homotopy-Smoothing Methods for Variational Inequalities , 1999 .

[30]  L. Qi.  Regular Pseudo-Smooth NCP and BVIP Functions and Globally and Quadratically Convergent Generalized Newton Methods for Complementarity and Variational Inequality Problems , 1999 .