A Regularization Newton Method for Solving Nonlinear Complementarity Problems

Abstract. In this paper we construct a regularization Newton method for solving the nonlinear complementarity problem (NCP(F )) and analyze its convergence properties under the assumption that F is a P0 -function. We prove that every accumulation point of the sequence of iterates is a solution of NCP(F ) and that the sequence of iterates is bounded if the solution set of NCP(F ) is nonempty and bounded. Moreover, if F is a monotone and Lipschitz continuous function, we prove that the sequence of iterates is bounded if and only if the solution set of NCP(F ) is nonempty by setting $t=\frac{1}{2}$ , where $t\in [\frac{1}{2},1]$ is a parameter. If NCP(F) has a locally unique solution and satisfies a nonsingularity condition, then the convergence rate is superlinear (quadratic) without strict complementarity conditions. At each step, we only solve a linear system of equations. Numerical results are provided and further applications to other problems are discussed.

[1]  E. H. Zarantonello.  Projections on Convex Sets in Hilbert Space and Spectral Theory: Part I. Projections on Convex Sets: Part II. Spectral Theory , 1971 .

[2]  R. Rockafellar.  Monotone Operators and the Proximal Point Algorithm , 1976 .

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

[4]  R. Rockafellar.  MONOTONE OPERATORS AND AUGMENTED LAGRANGIAN METHODS IN NONLINEAR PROGRAMMING , 1978 .

[5]  L. Watson.  Solving the Nonlinear Complementarity Problem by a Homotopy Method , 1979 .

[6]  Martin Grötschel,et al.  Mathematical Programming The State of the Art, XIth International Symposium on Mathematical Programming, Bonn, Germany, August 23-27, 1982 , 1983, ISMP.

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

[8]  Lars Mathiesen,et al.  An algorithm based on a sequence of linear complementarity problems applied to a walrasian equilibrium model: An example , 1987, Math. Program..

[9]  P. Subramanian.  A note on least two norm solutions of monotone complementarity problems , 1988 .

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

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

[12]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

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

[14]  T. Zolezzi,et al.  Well-Posed Optimization Problems , 1993 .

[15]  Jong-Shi Pang,et al.  NE/SQP: A robust algorithm for the nonlinear complementarity problem , 1993, Math. Program..

[16]  V. Venkateswaran,et al.  An Algorithm for the Linear Complementarity Problem with a $P_0 $-Matrix , 1993, SIAM J. Matrix Anal. Appl..

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

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

[19]  Patrick T. Harker,et al.  A nonsmooth Newton method for variational inequalities, II: Numerical results , 1994, Math. Program..

[20]  A. Fischer.  An NCP–Function and its Use for the Solution of Complementarity Problems , 1995 .

[21]  Masao Fukushima,et al.  Modified Newton methods for solving a semismooth reformulation of monotone complementarity problems , 1996, Math. Program..

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

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

[24]  Christian Kanzow,et al.  Global Convergence Properties of Some Iterative Methods for Linear Complementarity Problems , 1996, SIAM J. Optim..

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

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

[27]  Stephen J. Wright,et al.  A superlinear infeasible-interior-point algorithm for monotone complementarity problems , 1996 .

[28]  O. Nelles,et al.  An Introduction to Optimization , 1996, IEEE Antennas and Propagation Magazine.

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

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

[31]  Stephen J. Wright.  Primal-Dual Interior-Point Methods , 1997, Other Titles in Applied Mathematics.

[32]  Houyuan Jiang,et al.  A New Nonsmooth Equations Approach to Nonlinear Complementarity Problems , 1997 .

[33]  Paul Tseng,et al.  An Infeasible Path-Following Method for Monotone Complementarity Problems , 1997, SIAM J. Optim..

[34]  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..

[35]  Michael C. Ferris,et al.  Engineering and Economic Applications of Complementarity Problems , 1997, SIAM Rev..

[36]  Francisco Facchinei,et al.  Regularity Properties of a Semismooth Reformulation of Variational Inequalities , 1998, SIAM J. Optim..

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

[38]  M. Seetharama Gowda,et al.  Weak Univalence and Connectedness of Inverse Images of Continuous Functions , 1999, Math. Oper. Res..

[39]  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..

[40]  Keisuke Hotta,et al.  Global convergence of a class of non-interior point algorithms using Chen-Harker-Kanzow-Smale functions for nonlinear complementarity problems , 1999, Math. Program..

[41]  F. Facchinei,et al.  Beyond Monotonicity in Regularization Methods for Nonlinear Complementarity Problems , 1999 .

[42]  Michael C. Ferris,et al.  Smooth methods of multipliers for complementarity problems , 1999, Math. Program..

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

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

[45]  G. Duclos.  New York 1987 , 2000 .

[46]  Defeng Sun,et al.  Improving the convergence of non-interior point algorithms for nonlinear complementarity problems , 2000, Math. Comput..