A Hybrid Modified BFGS Method For Symmetric Nonlinear Equations

Nonlinear equation problem is a kind of problem with wild applications.Moreover.quasi-Newton algorithms are very popular methods for solving optimization problems and nonlinear equations,which are also efficient algorithms with locally super-linear convergent properties.Especially when dealing with nonlinear equations problems,by choosing suitable line search methods,we can get the global convergent properties of quasi-Newton algorithms.On solving symmetric nonlinear equations problems,Li-Fukushima proposed a Gauss-Newton based BFGS algorithm,and proved its global and super-linear convergent properties on certain conditions.The algorithm is approximately norm decent.Based on this,Gu-Li-Qi-Zhou proposed norm descent algorithms by combining two BFGS modified forms.The modifications of BFGS make it more applicable with good properties such as having no requirement on the assumption of convexity of the function.By analyzing the two BFGS modification forms,modified BFGS(MBFGS)and cautious BFGS(CBFGS),we find that they both modify the algorithms while also existing shortcomings.Considering this,we propose a hybrid BFGS modification form,which is a combination of the good properties of both MBFGS and CBFGS such as no convexity requirement of the function and ensuring that the quasi-Newton matrix {Bk} is positive definite.In this paper,we focus on the problem of symmetric nonlinear equations.Moreover,by making use of the hybrid BFGS modification forms and suitable line search methods,we construct hybrid modified BFGS algorithms for solving symmetric nonlinear equations.We also give proofs on the global convergence and super-linear convergence of the algorithms.We also make some numerical experiments on a certain problem and show that our algorithms is available.Moreover,when dealing with higher dimension and more complicated problems,the hybrid modified BFGS method performs more steadily.