Polynomial interior point algorithms for general LCPs

Linear Complementarity Problems (LCP s) belong to the class of NP-complete problems. Therefore we can not expect a polynomial time solution method for LCP s without requiring some special property of the matrix coefficient matrix. Our aim is to construct some interior point algorithms which, according to the duality theorem in EP form, gives a solution of the original problem or detects the lack of property P∗(κ̃) (with arbitrary large, but apriori fixed κ̃) and gives a polynomial size certificate of it in polynomial time (depending on parameter κ̃, the initial interior point and the dimension of the LCP ). We give the general idea of a modification of interior point algorithms and present three concrete methods: affine scaling, long-step path-following and predictor-corrector interior point algorithm.