Complementarity Problems Is A Kind Of Modified Generalized Quasi-newton Method

In real life, many problem’s mathematical model can be expressed into thecomplementarity problem, the complementarity problem closely contact with the NonlinearProgramming, Minimax, the Game Theory, the Fixed Point Theory and other branches. Theemergence of the complementarity problem arouse people’s interest, many people have beeninvolved in the study. The complementary problem is an important class of optimizationproblem. In practical applications, it appears in the field of the Engineering Physics, thebalance of economic and traffic; at the sanme time, it also appears in optimality conditions ofthe constrained optimization. Therefore, it is important to study. Since the complemetarityproblem is proposed, people have done series of research, propose a lot of efficient algorithms,more used methods are projection method, interior-point method, smooth(nonsmooth)Newton method, etc. In this paper, the complementrity problem is convert into unconstrainedoptimization by using Fischer-Burmerister function, then unconstrained optimization is solvedby modified generalized quasi-Newton algorithm. The improved algorithm has goodmumerical results verified by numerical experiments.The article is divided into three parts, the first chapter gives the nature and relatedtheorems of the complementary problem and the complementary function, and amended onthe basis of the Broyden family of quasi-Newton algorithm, obtain the article’s modifiedgeneralized quasi-Newton formula and deduce it’s inverse matrix.Given in Chapter II of this article correction step quasi-Newton algorithm and thealgorithm to prove the global convergence and local superlinear convergence of theamendment on the basis of this article revised generalized quasi-Newton algorithm,theoretically verified feasibility.Chapter Ⅲ on a classical nonlinear complementarity problem of numerical experimentsare tested to verify the algorithm is feasible and branches, and finally it is the algorithm ofMatlab and the M-file.Finally, through theoretical and numerical experiments to verify its feasibility, but thereare still some problems need to continue research and explore, such as correct thequasi-Newton formula to make the algorithm faster, explore the convergence conditions tomake the algorithm more general.