Newton Algorithm Based On Generalized Cauchy Equations Of Second Order Quasi Diagonal Intended

The unconstrained optimization problems exist in many important fieldsbroadly.We usually use descent algorithm to solve these problems,in this way wecan get the optimal value of problems by successive iteration.The Quasi-Newtonmethod is an effective method to solve the unconstrained optimizationproblems.In recent years,many scholars had studied and explored on theQuasi-Newton method,they put forward the diagonal sparse Quasi-Newtonmethod, the diagonal Quasi-Cauchy method and so on,and got a great manyimportant conclusion.Based on those academic achievement, the paper willpropose the extend second order Quasi-Cauchy equation and studied below.First,we introduced the basic knowledge,development situation and researchstatus of Quasi-Newton method.And presented the main work of this papersimply.The development of Quasi-Newton method is mature.For Quasi-Newtonmethod,the key is constructed a drop direction and line search rules.The dropdirection is depends on the Hessen matrix or its inverse matrix, Hessen matrix orits inverse is depends on the Quasi-Newton equation. Based on theQuasi-Cauchy equations,the article first proposed the generalized second orderQuasi-Cauchy equation.The new equation extended second order Quasi-Cauchyto a class of Quasi-Cauchy equation.Based on the new equation,we constructeda diagonal updating formula by using the least-change strategy,and proposedthree algorithms below.Based on the diagonal updating formula,we designed a new method forsolving the unconstrained optimization problems by introducing a simplemonotone strategy,that is,the monotone gradient method based on thegeneralized second order Quasi-Cauchy equation.Under certain conditions, weproved the global convergence and liner convergence of the method.Numericalresults show that the new method is effective and practical.One of the key steps in Quasi-Newton methods is line search technique.Forthe monotone gradient method based on the generalized second orderQuasi-Cauchy equation,we didn’t need line search to ensure convergence of the algorithm.Or it can been seen we selected special k1as the line length.In thelater two chapeters,based on the generalized second order Quasi-Cauchyequation,we constracted the generalized diagonal second order Quasi-Cauchymethod with Armijo line search and the generalized diagonal second orderQuasi-Cauchy method under a larger search step for solving unconstrainedoptimization problems by using Armijo unprecise line search and Nonmonooneline search to ensure search step.And proved the global convergence of thealgorithm.Numerical results show that the new methods are effective.