Sub-quadratic convergence of a smoothing Newton algorithm for the P(0) and monotone LCP
Given M is an element of R(nxn) and q is an element of R(n), the linear complementarity problem (LCP) is to find (x, s) is an element of R(n) x R(n) such that (x, s) greater than or equal to 0, s = M(x) + q, x(T)s = 0. By using the Chen-Harker-Kanzow-Sniale (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 [Matheniatical Programming, Vol. 87, 2000, pp. 1-35], is proposed to solve the LCP with M being assumed to be a P(0)-matrix (P(0)-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 P(0)-LCP: (ii) it generates a bounded sequence if the P(0)-LCP has a nonempty and bounded solution set: (iii) if an accumulation point of the iteration sequence satisfies a nonsingularity condition, which implies the P(0)-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 is an element of (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.
sub-quadratic convergence linear complementarity problem smoothing newton method global convergence