On the Finite Termination of an Entropy Function Based Non-Interior Continuation Method for Vertical Linear Complementarity Problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that under some milder than usual assumptions the proposed algorithm finds an exact solution of VLCP in a finite number of iterations. Some computational results are included to illustrate the potential of this approach.

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

[2]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[3]  Zheng-Hai Huang,et al.  Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem , 2003 .

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

[5]  Andreas Fischer,et al.  On finite termination of an iterative method for linear complementarity problems , 1996, Math. Program..

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

[7]  Chengjian Zhang,et al.  Stability analysis of Volterra delay-integro-differential equations and their backward differentiation time discretization , 2004 .

[8]  P. Tseng.  Error Bounds and Superlinear Convergence Analysis of Some Newton-Type Methods in Optimization , 2000 .

[9]  Yinyu Ye,et al.  On the finite convergence of interior-point algorithms for linear programming , 1992, Math. Program..

[10]  Zhenghai Huang,et al.  Predictor-Corrector Smoothing Newton Method, Based on a New Smoothing Function, for Solving the Nonlinear Complementarity Problem with a P0 Function , 2003 .

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

[12]  D. Bertsekas,et al.  A new penalty function method for constrained minimization , 1972, CDC 1972.

[13]  Sanjay Mehrotra,et al.  Finding an interior point in the optimal face of linear programs , 1993, Math. Program..

[14]  Li-Zhi Liao,et al.  Regularized Smoothing Approximations to Vertical Nonlinear Complementarity Problems , 1999 .

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

[16]  Xiaojun Chen,et al.  A Global Linear and Local Quadratic Continuation Smoothing Method for Variational Inequalities with Box Constraints , 2000, Comput. Optim. Appl..

[17]  D Sun.  ON THE FINITE TERMINATION OF THE DAMPED-NEWTON ALGORITHM FOR LINEAR COMPLEMENTARITY PROBLEM , 1998 .

[18]  S. Fang,et al.  Entropy Optimization and Mathematical Programming , 1997 .

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

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

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

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

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

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

[25]  P. Tseng.  Analysis Of A Non-Interior Continuation Method Based On Chen-Mangasarian Smoothing Functions For Com , 1998 .

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

[27]  Zhiming Chen,et al.  An Augmented Lagrangian Method for Identifying Discontinuous Parameters in Elliptic Systems , 1999 .

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

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

[30]  Shu-Cherng Fang,et al.  On the entropic regularization method for solving min-max problems with applications , 1997, Math. Methods Oper. Res..

[31]  E. S. Kuh,et al.  Piecewise-Linear Theory of Nonlinear Networks , 1972 .

[32]  Ji-Ming Peng,et al.  A non-interior continuation method for generalized linear complementarity problems , 1999, Math. Program..

[33]  Defeng Sun,et al.  Sub-quadratic convergence of a smoothing Newton algorithm for the P0– and monotone LCP , 2004, Math. Program..

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

[35]  Jiming Peng,et al.  A Strongly Polynomial Rounding Procedure Yielding a Maximally Complementary Solution for P*(kappa) Linear Complementarity Problems , 2000, SIAM J. Optim..

[36]  A. Hoffman.  On approximate solutions of systems of linear inequalities , 1952 .

[37]  Paul Tseng,et al.  On the convergence of the exponential multiplier method for convex programming , 1993, Math. Program..

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

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

[40]  Shu-Cherng Fang,et al.  On the entropic perturbation and exponential penalty methods for linear programming , 1996 .