Algorithms For Nonlinear Weighted Complementarity Problems

Complementarity problem is a hot research direction in the field of operations research.Due to its extensive application,it has attracted extensive attention and strong research interest from many scholars since it was first proposed in 1963.In the following decades,it has achieved fruitful results in both theoretical research and algorithm research fields of complementarity problems.In 2012,Florian A.Potra,an internationally renowned optimization expert,proposed the mathematical model of weighted complementarity problem,which greatly expanded the existing complementarity problem and has been widely applied in the fields of economy,management,engineering calculation,etc.,which has attracted extensive attention from researchers.By reading a lot of literature,on the basis of grasping the basic theory of complementarity problem and various algorithm ideas,this paper will study the nonlinear weighted complementarity problem,aiming to design smoothing algorithms for solving the nonlinear weighted complementarity problem,and analyze the global and local convergence properties of these algorithms.This paper is divided into the following four parts.In the first chapter,the research status and related basic knowledge of nonlinear weighted complementarity problem are introduced.In the second chapter,we study a monotone smoothing algorithm for nonlinear weighted complementarity problem.The algorithm uses a monotone line search method and a weighted smoothing function to transform the nonlinear weighted complementarity problem into a smoothing system of equations,which are then solved by Newton method.Under appropriate assumptions,we prove that the algorithm has global and local quadratic convergence properties.The numerical results show that the algorithm is very efficient.In the third chapter,we study a non-monotone smoothing algorithm for nonlinear weighted complementarity problems.Firstly,a new smoothing function is introduced,which can be used to reconstruct the nonlinear weighted complementarity problem into a smoothing nonlinear system.On this basis,we propose a new smoothing algorithm,which adopts the non-monotone line search method.In each iteration,the algorithm only needs to solve a linear system of equations and perform a non-monotone line search.Under appropriate assumptions,the global and local quadratic convergence of the algorithm is demonstrated.Numerical experimental results show that the proposed non-monotone smoothing algorithm is more effective than the existing monotone smoothing algorithm.In the fourth chapter,we summarize the content of the article.