The linear complementarity problem under asymptotic analysis

In this work we study the classical linear complementarity problem LCP by describing the asymptotic behavior of the approximate solutions to its variational inequality formulation. Thus, some properties satisfied by the directions which are limits of the normalized unbounded approximate solutions will be established. Based on this analysis, various equivalent conditions guaranteeing the existence of solutions to LCP are given. In particular, the sufficient condition of Gowda and Pang expressed in terms of the solutions to augmented linear complementarity problems is written in a way that is more easily verifiable. Our approach allows us to deal with Garcia-matrices, semimonotone, copositive, q-pseudomonotone matrices among others, in a unified framework. Furthermore, we introduce a larger class of matrices for which many of the results (including a sensitivity one) due to Gowda and Pang are still valid. In addition, some conditions ensuring the boundedness of the solution set are also provided, and some estimates for the asymptotic cone of the solution set, for different classes of matrices, are given as well. Hence, the present approach sheds new light and offers an alternative to view classical results.