A smoothing Newton method for extended vertical linear complementarity problems

In this paper, we reformulate the extended vertical linear complementarity problem (EVLCP(m, q)) as a nonsmooth equation H(t, x) = 0, where H : ℝ^n+l → ℝⁿ⁺¹, t ∈ ℝ is a parameter variable, and cursive Greek chi ∈ ℝ is the original variable. H is continuously differeritiable except at such points (t, cursive Greek chi) with t = 0. Furthermore H is strongly semismooth. The reformulation of EVLCP(m, q) as a nonsmooth equation is based on the so-called aggregation (smoothing) function. As a result, a Newton-type method is proposed which generates a sequence {w^k = (t^k,cursive Greek chi^k)} with all t^k > 0. We prove that every accumulation point of this sequence is a solution of EVLCP(M, q) under the assumption of row W₀-property. If row W-property holds at the solution point, then the convergence rate is quadratic. Promising numerical results are also presented.