Abstract
The focus of this article is to add a new class of rank one of modified Quasi-Newton techniques to solve the problem of unconstrained optimization by updating the inverse Hessian matrix with an update of rank 1, where a diagonal matrix is the first component of the next inverse Hessian approximation, The inverse Hessian matrix is generated by the method proposed which is symmetric and it satisfies the condition of modified quasi-Newton, so the global convergence is retained. In addition, it is positive definite that guarantees the existence of the minimizer at every iteration of the objective function. We use the program MATLAB to solve an algorithm function to introduce the feasibility of the proposed procedure. Various numerical examples are given`.
Highlights
There are many efforts to achieve a better approximation of the Hessian matrix
In [2], the authors find a revamped Broyden family composed of BFGS ( Broyden, Fletcher, Goldfard, andShanno), which analogs to the changing that suggested by authors in [1,3]
BFGS update was updated on the basis of the new Quasi-Newton condition where is a matrix
Summary
There are many efforts to achieve a better approximation of the Hessian matrix. The modified Quasi-Newton (secant) condition is proposed by Zhang J, and Xu Ch. [1]. In [5], the authors suggested the Broyden update to guarantee the positive definite property of Hessian matrix and to give the global convergence of the proposed process. In [10], the authors proposed the modified BFGS update (H-version) by updating the vector s ( solution-current solution) and they provided that the proposed method preserve the strong positive definite property and globally convergent. Even the Hessian matrix is not analytically available, or the Hessian's estimation is challenging These refer to a class of techniques that use only the values of the equation and the gradients of the objective function that are closely related to the method of Newton. The problem is to solve problem (1) by generating a symmetrical and positive definite inverse Hessian matrix sequence that satisfies the quasi-Newton condition at any iteration
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