Sub-quadratic convergence of a smoothing Newton algorithm for the P 0- and monotone LCP

Given M ∈ <n×n and q ∈ <, the linear complementarity problem (LCP) is to find (x, s) ∈ < × < such that (x, s) ≥ 0, s = Mx + q, x s = 0. By using the ChenHarker-Kanzow-Smale (CHKS) smoothing function, the LCP is reformulated as a system of parameterized smooth-nonsmooth equations. As a result, a smoothing Newton algorithm, which is a modified version of the Qi-Sun-Zhou algorithm [Mathematical Programming, Vol. 87, 2000, pp. 1–35], is proposed to solve the LCP with M being assumed to be a P0-matrix (P0−LCP). The proposed algorithm needs only to solve one system of linear equations and to do one line search at each iteration. It is proved in this paper that the proposed algorithm has the following convergence properties: (i) it is well-defined and any accumulation point of the iteration sequence is a solution of the P0−LCP; (ii) it generates a bounded sequence if the P0−LCP has a nonempty and bounded solution set; (iii) if an accumulation point of the iteration sequence satisfies a nonsingularity condition, which implies the P0−LCP has a unique solution, then the whole iteration sequence converges to this accumulation point sub-quadratically with a Q-rate 2 − t, where t ∈ (0, 1) is a parameter; and (iv) if M is positive semidefinite and an accumulation point of the iteration sequence satisfies a strict complementarity condition, then the whole sequence converges to the accumulation point quadratically.