Global error bounds for the extended vertical LCP

Abstract A new necessary and sufficient condition for the row $\mathcal{W}$ -property is given. By using this new condition and a special row rearrangement, we provide two global error bounds for the extended vertical linear complementarity problem under the row $\mathcal{W}$ -property, which extend the error bounds given in Chen and Xiang (Math. Program. 106:513–525, 2006) and Mathias and Pang (Linear Algebra Appl. 132:123–136, 1990) for the P-matrix linear complementarity problem, respectively. We show that one of the new error bounds is sharper than the other, and it can be computed easily for some special class of the row $\mathcal{W}$ -property block matrix. Numerical examples are given to illustrate the error bounds.

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

[2]  Xiaojun Chen,et al.  Computation of Error Bounds for P-matrix Linear Complementarity Problems , 2006, Math. Program..

[3]  Olvi L. Mangasarian,et al.  New improved error bounds for the linear complementarity problem , 1994, Math. Program..

[4]  J. J. Moré,et al.  Smoothing of mixed complementarity problems , 1995 .

[5]  Xiaojun Chen,et al.  Computation of generalized differentials in nonlinear complementarity problems , 2011, Comput. Optim. Appl..

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

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

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

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

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

[11]  J. Rohn.  Systems of linear interval equations , 1989 .

[12]  Naihua Xiu,et al.  A characteristic quantity of P-matrices , 2002, Appl. Math. Lett..

[13]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[14]  Michael C. Ferris,et al.  Complementarity and variational problems : state of the art , 1997 .

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

[16]  Richard W. Cottle,et al.  Linear Complementarity Problem , 2009, Encyclopedia of Optimization.

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

[18]  J. Pang,et al.  Error bounds for the linear complementarity problem with a P-matrix , 1990 .