Global linear and quadratic one-step smoothing newton method for vertical linear complementarity problems

A one-step smoothing Newton method is proposed for solving the vertical linear complementarity problem based on the so-called aggregation function. The proposed algorithm has the following good features: (i) It solves only one linear system of equations and does only one line search at each iteration; (ii) It is well-defined for the vertical linear complementarity problem with vertical block P ₀ matrix and any accumulation point of iteration sequence is its solution. Moreover, the iteration sequence is bounded for the vertical linear complementarity problem with vertical block P ₀ +R ₀ matrix; (iii) It has both global linear and local quadratic convergence without strict complementarity. Many existing smoothing Newton methods do not have the property (iii). Foundation items: the National Natural Science Foundation of China (10201001); the National Outstanding Young Investigator Grant (70225005) Biography: Zuphang Li-ping (1970≈)