A Modified Two-parameter Broyden Family Algorithm

Unconstrained optimization problems widely exist in engineering design,image processing and other fields,the study of such problem is of great significance in both theory and practical application.Quasi-Newton method is a classical method for solving unconstrained optimization problems.The Broyden family method is one of the most effective quasi-Newton methods for solving unconstrained optimization problems.Due to the complex structure of the Broyden family formula,the study of the global convergence of the Broyden family method is not enough.It is well known that the iterative matrix in quasi-Newton method is an approximation of the Hessian matrix of the objective function.Then,the search direction is generated based on the Newton equation,and the iteration is complete with the line search.Therefore,the properties of iterative matrix in quasi-Newton method greatly affect the efficiency of the algorithm.In this thesis,a class of modified Broyden family method is proposed based on the classical Broyden family method,which reduces the influence of eigenvalues of iterative matrix on the algorithm and improves the efficiency of the algorithm.First,some research background of unconstrained optimization,some recent research status at home and abroad,and some basic concepts of optimization are introduced in this dissertation.Second,based on the classical Broyden family method,a two-parameter Broyden family method with the projection technique is designed.Under certain assumptions,the global convergence of the method is obtained.At the end of the chapter,the proposed method is compared with the classical Broyden family method,and the results show that the two-parameter Broyden family method has better numerical results.In addition,considering the need to store a large number of iterative matrices,the iterative matrix Bk of the two-parameter Broyden family formula is replaced by the constant times of the identity matrix,and the two-parameter memoryless Broyden family method is designed.Under appropriate assumptions,the global convergence of the method is proved.Compared with the experimental results,the two parameter memoryless Broyden family method is more efficient.