Some Properties of Generalized Positive Subdefinite Matrices

Abstract

The class of generalized positive subdefinite (GPSBD) matrices is an interesting matrix class introduced by Crouzeix and Komlósi [Appl. Optim. 59, Kluwer, Dordrecht, The Netherlands, 2001, pp. 45-63]. In this paper, we obtain some properties of GPSBD matrices. We show that copositive GPSBD matrices are $P_{0}$ and a merely generalized positive subdefinite (MGPSBD) matrix with some additional conditions belongs to the class of row sufficient matrices introduced by Cottle, Pang, and Venkateswarn [Linear Algebra Appl., 114/115 (1989), pp. 231-249]. Further, it is shown that for a subclass of GPSBD matrices, the solution set of a linear complementarity problem is same as the set of Karush--Kuhn--Tucker-stationary points of the corresponding quadratic programming problem. We provide a counter example to show that a copositive GPSBD matrix need not be sufficient in general. Finally, we show that if a matrix A can be written as a sum of a copositive-plus MGPSBD matrix with an additional condition and a copositive matrix and if it satisfies a feasibility condition, then Lemke's algorithm can solve LCP$(q,A).$ This further extends the applicability of Lemke's algorithm and a result of Evers.