The Global Linear Convergence of a Noninterior Path-Following Algorithm for Linear Complementarity Problems

Abstract

A noninterior path-following algorithm is proposed for the linear complementarity problem. The method employs smoothing techniques introduced by Kanzow. If the LCP is P 0 / R 0 and satisfies a nondegeneracy condition due to Fukushima, Luo, and Pang, then the algorithm is globally linearly convergent. As with interior point path-following methods, the convergence theory relies on the notion of a neighborhood for the central path. However, the choice of neighborhood differs significantly from that which appears in the interior point literature. Numerical experiments are presented that illustrate the significance of the neighborhood concept for this class of methods.

Existence and Limiting Behavior of a Non--Interior-Point Trajectory for Nonlinear Complementarity Problems Without Strict Feasibility Condition

“…Other non-interior-point algorithms in the literature also suffer from the same restriction. For instance, the algorithms developed by Chen and Harker [5], Burke and Xu [1], and Chen and Chen [6] require the P 0 and R 0 assumption, which also implies that the solution set of the CP is nonempty and bounded, and hence the problem is strictly feasible. The strict feasibility condition plays an indispensable role in these known non-interior-point methods.…”

“…Due to the impressive numerical performance of the algorithm as well as its ideal convenience for the application to those CPs where interiority restriction on the iterates is quite severe, there are a growing interest and fruitful results in the non-interior-point methods for the CP, see, e.g., Kanzow [18], Burke and Xu [1,2,3,4], Xu [35], Xu and Burke [36], Chen and Chen [6], Hotta and Yoshise [16], Hotta, Inaba, and Yoshise [17], Qi and Sun [26], and Tseng [32]. In the setting of P 0 -CPs, a common feature of the above-mentioned non-interior-point methods is to make an assumption of the strict feasibility condition (or the nonemptyness and the boundedness conditions on the solution set) and a proper condition.…”

A Globally and Locally Superlinearly Convergent Non--Interior-Point Algorithm for P0LCPs

“…However, no rate of convergence was reported in these papers. The first global linear convergence result for the LCP with a P 0 and R 0 matrix was obtained by Burke and Xu [3], who also proposed in [4] a non-interior-point predictor-corrector algorithm for monotone LCPs which was both globally linearly and locally quadratically convergent under certain assumption. Further development of non-interior-point methods can be found in [35,5,40,33,8,7,21].…”

“…For P 0 LCPs, it is shown (see [42,43]) that most assumptions used for non-interior-point algorithms, for instance, the Condition 1.5 in [25], Condition 1.2 in Hotta and Yoshise [20], and the P 0 + R 0 assumption in Burke and Xu [3] and Chen and Chen [7], imply that the solution set of the problem is bounded. As showed by Ravindran and Gowda in [34] the P 0 complementarity problem with a bounded solution set must have a strictly feasible point, i.e., there exists an x 0 such that M x 0 + d > 0.…”

“…c) Assumption 2.2 of Chen and Chen [7]. d) f is a P 0 and R 0 function [3,7]. e) f is a P * -function (i.e., M is a P * matrix) and there is a strictly feasible point [26].…”