A Smoothing Method For Solving Vertical Complementarity Problem

In this thesis,a smoothing method for solving vertical linear complementarity problem and vertical nonlinear complementarity problem is given based on the quadratic positive sign function.Firstly,the vertical linear complementarity problem and vertical nonlinear complementarity problem are transformed into an equivalent the nonsmoothed system of equations with the maximum function;secondly,based on the quadratic positive sign function,a smoothing mapping of the equivalent nonsmoothing system is constructed,and the variables of the mapping include independent variables,auxiliary variables and smoothing parameters;finally,the smoothing parameter in the smoothing map is made into continuous variables,and the smoothing map for solving vertical linear complementarity problems and vertical nonlinear complementarity problems are constructed.For the vertical linear complementarity problem,when the traditional smoothing algorithm is used to solve the smoothing equation,the sum of the quadratic positive sign function of the component function of the smoothing map cound be 0,resulting in the singularity of the Jacobian matrix of the smoothing map,resulting in the failure of the algorithm.In this thesis,a strategy is proposed to adjust the auxiliary variables and parameters so that the minimum value in the component of the adjusted smoothing map is greater than or equal to the 1 norm of the smoothing map.Combined with the traditional smoothing algorithm,an improved smoothing algorithm is given.Because the improved smoothing algorithm can only obtain an approximate solution of the vertical linear complementarity problem.In order to improve the computational efficiency,this thesis combines the improved smoothing algorithm with the Newton method for solving the exact solution of the vertical linear complementarity problem,and gives a smoothing algorithm for solving the vertical linear complementarity problem.The numerical experimental results show that the algorithm has high computational efficiency.Based on the smoothing map constructed by the vertical linear complementarity problem,a smoothing map for solving the vertical nonlinear complementarity problem is constructed,and the smoothing algorithm for solving the smoothing map of the vertical linear complementarity problem is used to solve the smoothing map of the vertical nonlinear complementarity problem.This thesis,the exact solution of the vertical nonlinear complementarity problem is obtained by Newton method form the approximate solution.It is showed that the algorithm is feasible and efficient by numerical experiments.