A Non-Interior Predictor-Corrector Path-Following Method for LCP

Abstract

In a previous work the authors introduced a non—interior predictor-corrector path following algorithm for the monotone linear complementarity problem. The method uses Chen—Harker—Kanzow—Smale smoothing techniques to track the central path and employs a refined notion for the neighborhood of the central path to obtain the boundedness of the iterates under the assumption of monotonicity and the existence of a feasible interior point. With these assumptions, the method is shown to be both globally linearly convergent and locally quadratically convergent. In this paper it is shown that this basic approach is still valid without the monotonicity assumption and regardless of the choice of norm in the definition of the neighborhood of the central path. Furthermore, it is shown that the method can be modified so that only one system of linear equations needs to be solved at each iteration without sacrificing either the global or local convergence behavior of the method. The local behavior of the method is further illuminated by showing that the solution set always lies in the interior of the neighborhood of the central path relative to the affine constraint. In this regard, the method is fundamentally different from interior point strategies where the solution set necessarily lies on the boundary of the neighborhood of the central path relative to the affine constraint. Finally, we show that the algorithm is globally convergent under a relatively mild condition.