The linear complementarity problem with a parametric input

The linear complementarity problem (LCP) and its importance is well known. Over the past decade, an increasing number of studies have focused on the theory of sensitivity analysis which discusses a local perturbation behavior of current solutions in response to small disturbances of data. There are two questions, however, that remain unclear: (i) What condition, if any, results in changes of nonzero variable sets? (ii) If a nonzero variable set stays the same for certain perturbations, can LCP solutions be expressed as a function of the current solution and perturbed parameters? In this paper, we examine the two issues for the LCP with a single parametric row and general random matrices. The outcome of this paper significantly improves the results obtained in previous research. In particular, we establish the conditions which ensure the convexity of basis set and uniqueness of solution basis in both situations.