The Linear Complementarity Problem, Sufficient Matrices, and the Criss-Cross Method

Specially structured linear complementarity problems (LCPs) and their solution by the miss-cross method are examined. The criss-cross method is known to be finite for LCPs 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 Pand PSD matrices. Cottle also showed that the principal pivoting method (with a clever modification) can be applied to row sufficient LCPs. In this paper the finiteness of the criss-cross method for sufficient LCPs is proved. Further it is shown that a matrix is sufficient if and only if the miss-cross method processes all the LCPs defined by this matrix and all the LCPs defined by the transpose of this matrix and any parameter vector.