On generalizations of positive subdefinite matrices and the linear complementarity problem

In this paper, we introduce the notion of generalized positive subdefinite matrices of level k by generalizing the definition of generalized positive subdefinite matrices introduced by Crouzeix and Komlosi in [ Applied optimization. Vol. 59, Dordrecht: Kluwer Academic Publications; 2001. p. 45-63]. The main motivation behind this generalization is to identify new subclasses of row sufficient matrices [ Cottle, Pang and Venkateswaran, Linear Algebra Appl. 1989; 114/ 115: 231-249] which are not a subclass of copositive matrices. We establish new sufficient conditions for a matrix to be a non-copositive P0 matrix and a non-copositive row sufficient matrix using the notion of generalized positive subdefinite matrices of level k. We present a new termination result for Lemke's algorithm which supplements Jones' result [ Math Program. 1986; 35: 239-242].