The Broyden-type Methods For General And Generalized Nonlinear Complementarity Problems

Complementary problems,arising from the differential equation,the game theory,engineering problems and transportation,have been received many attentions,and developed into a hot research topic of nonlinear optimization.This article focus on the theory and method for nonlinear complementarity problem(NCP)and generalized nonlinear complementarity problem(GNCP).In solving nonlinear complementarity problem,an independent variable is used to strictly approximate the function ().And then the NCP is transformed into the equivalent equations by 3-1 piecewise NCP function.Based on the derivative definition of 3-1 piecewise NCP function,we present Newton's method and Broyden-like method together with modified nonmonotone technique for NCP.In Newton's method,only one system of equations and a nonmonotone line search are needed to get the trial point at each iteration.In Broyden-like method,we present the update formula for coefficient matrix of the equations so that decrease the computational account,and then obtain the search direction by the nonmonotone line search.For generalized nonlinear complementarity problem,it is similar to the nonlinear complementarity problem.We adopt the independent variables to approximate the function ()and ().Based on the 3-1 piecewise NCP function,we propose Newton's method,Broyden-like method and mixed method for GNCP.In the above algorithms,only one equation and one nonmonotone line search are needed at each iteration.Furthermore,we give the update formula of independent variables,which are the approximation to the function ()and ()in GNCP.Compared to the existing methods,our methods are more flexible and easier to carry out,and the amount of computation is smaller.Under the reasonable conditions,we prove the global convergence properties of the proposed methods,and give some numerical results to show their effectiveness.