On generalizations of positive subdefinite matrices and the linear complementarity problem

TLDR: The notion of generalized positive subdefinite matrices of level k is introduced by generalizing the definition of generalized Positive Subdefinite Matrices introduced by Crouzeix and Komlósi in [Applied optimization].

Abstract: 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 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].