A non-interior continuation method for generalized linear complementarity problems

Abstract.In this paper, we propose a non-interior continuation method for solving generalized linear complementarity problems (GLCP) introduced by Cottle and Dantzig. The method is based on a smoothing function derived from the exponential penalty function first introduced by Kort and Bertsekas for constrained minimization. This smoothing function can also be viewed as a natural extension of Chen-Mangasarian’s neural network smooth function. By using the smoothing function, we approximate GLCP as a family of parameterized smooth equations. An algorithm is presented to follow the smoothing path. Under suitable assumptions, it is shown that the algorithm is globally convergent and local Q-quadratically convergent. Few preliminary numerical results are also reported.

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

[2]  Roman Sznajder,et al.  A generalization of the Nash equilibrium theorem on bimatrix games , 1996 .

[3]  M. Kostreva,et al.  Global solvability of generalized linear complementarity problems and a related class of polynomial complementarity problems , 1992 .

[4]  R. Sznajder,et al.  The Generalized Order Linear Complementarity Problem , 1994, SIAM J. Matrix Anal. Appl..

[5]  Li-Zhi Liao,et al.  A Smoothing Newton Method for Extended Vertical Linear Complementarity Problems , 1999, SIAM J. Matrix Anal. Appl..

[6]  Robert M. Freund,et al.  Interior point methods : current status and future directions , 1996 .

[7]  L. Vandenberghe,et al.  The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits , 1989, IEEE International Symposium on Circuits and Systems,.

[8]  Shinji Mizuno,et al.  A General Framework of Continuation Methods for Complementarity Problems , 1993, Math. Oper. Res..

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

[10]  D. Bertsekas.  Approximation procedures based on the method of multipliers , 1977 .

[11]  唐焕文,et al.  A maximum entropy method for convex programming , 1995 .

[12]  Michael M. Kostreva,et al.  The generalized Leontief input-output model and its application to the choice of new technology , 1993, Ann. Oper. Res..

[13]  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..

[14]  S. M. Robinson.  Some continuity properties of polyhedral multifunctions , 1981 .

[15]  Eugene L. Allgower,et al.  Numerical continuation methods - an introduction , 1990, Springer series in computational mathematics.

[16]  James V. Burke,et al.  The Global Linear Convergence of a Noninterior Path-Following Algorithm for Linear Complementarity Problems , 1998, Math. Oper. Res..

[17]  Jinlin Peng,et al.  A Smoothing Function and Its Applications , 1998 .

[18]  Nimrod Megiddo,et al.  Homotopy Continuation Methods for Nonlinear Complementarity Problems , 1991, Math. Oper. Res..

[19]  Min Sun,et al.  Monotonicity of Mangasarian's iterative algorithm for generalized linear complementarity problems , 1989 .

[20]  Dimitri P. Bertsekas,et al.  A new algorithm for solution of resistive networks involving diodes , 1976 .

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

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

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

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

[25]  S. Min.  Singular control problems in bounded intervals , 1987 .

[26]  G. Dantzig,et al.  A generalization of the linear complementarity problem , 1970 .

[27]  A. Fischer.  A Newton-type method for positive-semidefinite linear complementarity problems , 1995 .

[28]  R. Sznajder,et al.  Generalizations of P0- and P-properties; extended vertical and horizontal linear complementarity problems , 1995 .

[29]  Xing-Si Li,et al.  AN AGGREGATE FUNCTION METHOD FOR NONLINEAR PROGRAMMING , 1991 .

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

[31]  Aniekan Ebiefung,et al.  Nonlinear mappings associated with the generalized linear complementarity problem , 1995, Math. Program..

[32]  Olvi L. Mangasarian,et al.  Smoothing methods for convex inequalities and linear complementarity problems , 1995, Math. Program..

[33]  Song Xu,et al.  The global linear convergence of an infeasible non-interior path-following algorithm for complementarity problems with uniform P-functions , 2000, Math. Program..

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

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

[36]  Bintong Chen,et al.  A Global Linear and Local Quadratic Noninterior Continuation Method for Nonlinear Complementarity Problems Based on Chen-Mangasarian Smoothing Functions , 1999, SIAM J. Optim..

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

[38]  Christian Kanzow,et al.  On the resolution of monotone complementarity problems , 1996, Comput. Optim. Appl..

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

[40]  R. Sridhar,et al.  The generalized linear complementarity problem revisited , 1996, Math. Program..