## The Linear Complementarity Problem, Sufficient Matrices and the Criss-Cross Method (1990)

Citations: | 6 - 4 self |

### BibTeX

@MISC{Hertog90thelinear,

author = {D. Den Hertog and C. Roos and T. Terlaky},

title = {The Linear Complementarity Problem, Sufficient Matrices and the Criss-Cross Method},

year = {1990}

}

### OpenURL

### Abstract

Specially structured Linear Complementarity Problems (LCP's) and their solution by the criss-cross method are examined in this paper. The criss-cross method is known to be finite for LCP's with positive semidefinite bisymmetric matrices and with P-matrices. It is also a simple finite algorithm for oriented matroid programming problems. Recently Cottle, Pang and Venkateswaran identified the class of (column, row) sufficient matrices. They showed that sufficient matrices are a common generalization of P- and PSD-matrices. Cottle also showed that the principal pivoting method (with a clever modification) can be applied to row sufficient LCP's. In this paper the finiteness of the criss-cross method for sufficient LCP's is proved. Further it is shown that a matrix is sufficient if and only if the criss-cross method processes all the LCP's defined by this matrix and all the LCP's defined by the transpose of this matrix and any parameter vector.