Studies On Smoothing-type Methods For Weighted Linear Complementarity Problems

In 2012,Professor Florian A.Potra,an internationally famous optimization expert,proposed the weighted Linear Complementarity Problem(denoted by w LCP).The w LCP has wide applications in economy,management,atmospheric chemistry and multibody dynamics.In recent years,it has become a popular research topic in mathematical programming.Many researchers have studied the w LCP from theory and algorithm and have obtained many excellent results.This paper mainly studies the smooth-type algorithms for solving the w LCP and has obtained the following results:1.We propose a one-parametric class of smoothing functions and analyze its properties.By applying these smoothing functions to reformulate the w LCP as a system of smooth equations,we propose a new smoothing Newton method to solve it.Under the assumption that the w LCP is monotone,we prove that the method has global and local quadratic convergence.Different with current smoothing Newtontype methods,our method uses a nonmonotone derivative-free line search technique to generate the step size which makes it have better convergence properties and practical calculation effects.2.By using a complementarity function which is continuously differentiable everywhere,we reformulate the w LCP as a nonlinear equation and propose LevenbergMarquardt method to solve it.We show that the method is globally convergent without requiring the w LCP be monotone.Moreover,we prove that the method has local quadratic convergence rate under the local error bound condition which is weaker than the nonsingularity condition.Numerical results show that our method is very effective for solving monotone and nonmonotone w LCPs.3.We study a predictor-corrector smoothing Newton method for solving the special w LCP.Under suitable conditions,we prove that the method has global and local quadratic convergence.Specially,we prove that the merit function sequence converges to zero when the solution set is nonempty.Numerical experiments demonstrate that our method is very effective.