A Smoothing-type Algorithm For Solving The Weighted Complementarity Problem

In the field of operations research and applied mathematics,the complementarity problem has always been an active research direction.As an extension of the complementarity problem,the weighted complementarity problem has been widely used.For example,the equilibrium model in economics can be represented by a weighted complementary model.The purpose of this paper is to design a smoothing algorithm for solving the weighted complementarity problem,and to discuss the properties of the algorithm and the iteration generated by the algorithm.we introduce the research status of the problem at home and abroad,and reformulation method?smooth approximation function in non-linear complementarity problem.We propose corresponding designing ideas and specific steps of the algorithm.We also introduce two lemmas,and one of those is to prove that the smoothing algorithm is well-defined and the properties of the relevant sequence generated by the algorithm.Through these two lemmas,we prove that the boundness properties of the sequence generated by the algorithm and the algorithm is globally convergent.Then we prove that the solution generated by the algorithm is a maximally complementary solution of the weighted complementarity problem.Finally,we realize the smoothing algorithm in MATLAB.In the case of randomly selecting the data size,the algorithm can be solved successfully in a very small number of iterations and a very short computation time.The numerical results of the algorithm show that the proposed algorithm is effective.The work in this paper lays a foundation for perfecting the research of the complementarity problem and designing more algorithms to solve the weighted complementarity problem.