On the Solution Sets of Linear Complementarity Problems

In this article we consider two problems related to the solution sets of linear complementarity problems (LCPs)---one on the connectedness and the other on the convexity. In Jones and Gowda [ Linear Algebra Appl., 246 (1996), pp. 299--312], it was shown that the solution sets of LCPs arising out of $P_0 \cap Q$-matrices are connected, and they conjectured that this is true even in the case of $P_0\ \cap\ Q_0$-matrices. We verify this, at least in the case of nonnegative matrices. Our second problem is related to the class of fully copositive $(C_0^f)$-matrices introduced in Murthy and Parthasarathy [ Math. Programming, 82 (1998), pp. 401--411]. The class $C_0^f\cap Q_0,$ which contains the class of positive semidefinite matrices, has several properties that positive semidefinite matrices have. This article further supplements this by showing that the solution sets arising from LCPs with $C_0^f\cap Q_0$-matrices and their transposes are convex. This means that $C_0^f\cap Q_0$-matrices are sufficient matrices, another well known class in the theory of linear complementarity problem introduced by Cottle, Pang, and Venkateswaran [ Linear Algebra Appl., 114/115 (1989), pp. 231--249].