More on Scarf’s Complementarity Problem and its Error Bounds

In this paper, we revisit Scarf’s complementarity problem involving a rectangular matrix, known as the vertical block matrix introduced by [Cottle, RW and GB Dantzig (1970). A generalization of the linear complementarity problem. Journal of Combinatorial Theory, 8, 79–90]. We prove that the problem has a unique solution if the underlying matrix is a vertical block [Formula: see text]-matrix. Some results on the error bounds for the vertical block [Formula: see text]- and [Formula: see text]-matrices are derived. This further extends the results on global error bounds obtained by [Mathias, R and JS Pang (1990). Error bounds for the linear complementarity problem with a P-matrix. Linear Algebra and its Applications, 132, 123–136] and [Mangasarian, OL and J Ren (1994). New improved error bounds for the linear complementarity problem. Mathematical Programming, 66, 241–255].