A N EW M ODIFICATION I NTO S ELF -S CALING BFGS

In this paper, we propose a modified version of the self-scaling Yuan's [14] update which is based on the simple idea of approximation the objective function by technique is induced. Arithmetical Performance signifies that the new proposed techniques are more well-organized than the ordinary BFGS-technique.


Introduction :
Variable metric methods are a broadly utilized category from iterative techniques for solving the problem : where the step length k  satisfies the Wolfe conditioned : More details can be found in Fletcher [7,12]. The direction k d in ) 2 ( is that the resolution of the subsequent quadratic sub drawback : More details can be found in Ladislav [9]. For more studies and recent references on the variable metric, the interested researcher may refer to [1][2][3][4][5].
Below, I will now modify the derivation of Yuan's self-scaling method and study its convergence.

Metric
Based on the idea of Yuan's, we present a new Modification self-scaling variable metric.
The idea of Yuan's [14], important idea to derive the modified BFGS update as follows :

The equation
) 12 ( can be rewritten as : Note that the updates (12) and the usual BFGS updates are very different.
In fact, equation (13) with an exact line search 0 can be rearranged as the form : The above equation can be rewritten as : However, condition (15) may be modified further to give :

Now, equation
) 17 ( can be rewritten as : He showed that a modified QN-algorithm could be written as follows :

Global convergence analysis
We reading the convergence of the new methods with the normal states is needed. The Hessian matrix off is indicated by G .  (2) The Hessian matrix ) (x G is bounded above in norm for all . More details can be found in [8].
In [

Arithmetical Performance
In this section, arithmetical Performance is reported. We tested the variable metric algorithms with the following k  . We used test problems from Mor´e, Garbow, and Hillstrom [11]. They are summarized in Table  1 was satisfied, the program will be stopped. The following Himmeblau stop rule is used [13]. The program was also terminated if the number of iterations exceeds 1000".

Conclusions
In this paper, we have derived a new self-scaling variable metric following Yuan's Self-scaling [14]. This is a modification of the Yuan's formula. We find that this method performed better than the original BFGS method in each table, especially in Tables 1 with has been shown to be globally convergent.