Limiting Behavior of the Derivatives of Certain Trajectories Associated with a Monotone Horizontal Linear Complementarity Problem

Given a certain monotone horizontal linear complementarity problem (HLCP), we can naturally construct a family of systems of nonlinear equations parametrized by a parameter t ∈ (0, 1] with the property that, as t tends to 0, the corresponding system “converges” to the HLCP. Under reasonable conditions, it has been shown that each system of the family has a unique solution and that, as t tends to 0, these solutions converge to a specific solution of the HLCP. The main purpose of this paper is to study the asymptotic behavior of the derivative of the trajectory of solutions and therefore obtain information on the way the trajectory approaches the solution set of the HLCP. We show that the trajectory of solutions converges to the solution set along a unique and well-characterized direction. Moreover, if the HLCP has a solution satisfying strict complementarity then the direction forms a definite angle with any face of the feasible region which contains the limit point; otherwise, the direction is tangent to some face of the feasible region.