New criss-cross type algorithms for linear complementarity problems with sufficient matrices
We generalize new criss-cross type algorithms for linear complementarity problems (LCPs) given with sufficient matrices. Most LCP solvers require a priori information about the input matrix. The sufficiency of a matrix is hard to be checked ( no polynomial time method is known). Our algorithm is similar to Zhang's linear programming and Akkele, s, Balogh and Illes's criss-cross type algorithm for LCP-QP problems. We modify our basic algorithm in such away that it can start with any matrix M, without having any information about the properties of the matrix ( sufficiency, bisymmetry, positive definiteness, etc.) in advance. Even in this case, our algorithm terminates with one of the following cases in a finite number of steps: it solves the LCP problem, it solves its dual problem or it gives a certificate that the input matrix is not sufficient, thus cycling can occur. Although our algorithm is more general than that of Akkele, s, Balogh and Illes's, the finiteness proof has been simplified. Furthermore, the finiteness proof of our algorithm gives a new, constructive proof to Fukuda and Terlaky's LCP duality theorem as well.
linear complementarity problem sufficient matrix criss-cross algorithm alternative and ep theorems