A Rational Triangle Function as a Model for a Conjugate Gradient Optimization Method

College of Computer Sciences and Mathematics University of Mosul, Iraq Received on: 12/10/2003 Accepted on: 22/03/2004 ABSTRACT This paper presents the development and implementation of a new numberical based on a non-quadratic Triangular rational function model. For solving non-linear optimization problem .The algorithm is implemented in one version, employing exact line search. This version is compared numberically against versions of the CG-method. The results indicate that in general the new algorithm is superior to the previon algorithm.


Introduction
A more general model than the quadratic one is proposed in this paper as a basis for a CG algorithm. If q(x) is a quadratic function, then a function f is defined as a non-linear scaling of q(x) if the following condition holds : f = F(q(x)), dF/dq = F΄ > o and q(x) > 0 ………………. (1) where x* is the minimizer of q(x) with respect to x [13] .
The following properties are immediately derived from the above condition: i) Every contour line to q(x) is a contour line of f . ii) If x* is a minimzer of q(x), then it is a minimizer of f. iii) That x* is a global minimum of q(x) does not necessarily mean that it is a global minimum of f [5].

Various authors have puplished-related work in the area:
A conjugate method which minimizers the function f(x) = (q(x))  , and x  R n in at most step has been described by Fried [9]. Another special case, namely Where1 and 2 are scalars, has been investigated by Boland et al, [5].
Another model has been developed by Tassopoulos and Storey, [14] as follows: F(q(x) = 1 q(x) + 1/2q(x): 2 > 0 AL-Assady in [3] developed a model as follows :(F(q(x)) = In (q(x)) Al-Bayat, [1] has developed a new rational model which is defined as follows: F(q(x)) = 1 q(x)/1-2 q(x). Also Al-Bayati [4] developed an extended CG algorithm which is based on a general logarithmic model F(q(x) = log(q(x) -1 ) , > 0 And Al-Assady, [2] described there ECG algorithm which is based on the natural log function for the rational q(x) function F(q) = log , 2 < 0 In this paper, a new sine model is investigated and tested on a set of standard test function, on the assumed that condition (1) holds. An extended conjugate gradient algorithm is developed which is based on this new model which scales q(x) by the natural sinh function for the rational q(x) functions.

2.Theorem
Given an identical starting point x1,the method of Fletcher and Reeves [8]defined by and applied to f(q(x)) generate identical conjugate directions (within a positive multiple i f  ) and the identical sequence of approximations xi to the solution x* for any function satisfying (1).
It is assumed that the one-dimensional searches are exact. The vectors n i g g , 1 are gradients of f(q(x)) at x1 and x i ,respectively.

Proof:
The theorem is true For i=1, because Assume that , for i 2  , The implementation of the extended CG method has been performed for general function F(q(x) of the form of equations (2).
The unknown quantities i  were expressed in terms of available quantities of the algorithm.
The new model can now be written as Solving equation (2) for q And using the expression for from the above equation we have

In terms of the known quantities such a function and gradient values, from
Where Q is the Hessian Matrix and x * is the minimum point, we have:

Furthermore
Since therefore, we can express i  as follows:    (7) and (8)

4.The Outlines of our New Algorithm Area:
Given x0  R n an initial estimate of the minimizer x * .
Compute xi = xi-1 + i-1 di-1  Where i-1 is the optimal step size obtained by the line search procedure.
Step (3) : compute Where the derivation of scaling i  will be presented below.
Step (4)  Conjugate gradient methods are usually implemented by restarts in order to avoid an accumulation of errors affecting the search directions.
It is therefore generally agreed that restarting is very helpful in practices, so we have used the following restarting criterion in our practical investigations. If the new direction satisfies: Then a restart is also initiated. This new direction is sufficiently downhill in Powell [12]. The same line search was employed for all the methods. This was the cubic interpolation procedure described in Bunday [6].

The Numerical Experiments:
It is found that the NEW method which modifies CG-algorithm is better than the previous algorithm shown in Tables (1) and (2).