Page 32 - IJOCTA-15-1

S. Ben Hanachi, B. Sellami, M. Belloufi / IJOCTA, Vol.15, No.1, pp.25-34 (2025)
1964, Fletcher and Reeves 8 extended the CG convergence properties. However, they may not
method for solving nonlinear unconstrained min- perform well in practice due to jamming. On the
imization problems. The basis of all CG methods other hand, although HS and LS methods may
is to generate a sequence {x k } starting from an not converge in general, they often perform bet-
n
initial estimate x 0 ∈ R , using the following re- ter. To achieve good computational performance
currence: while maintaining strong global convergence, re-
searchers paid special attention to hybridize the
x k+1 = x k + α k d k , k = 0, 1, 2, . . . . (2) CG methods of the two categories. Various hy-
brid methods have been proposed in this context.
For example:
The step size α k is determined using a one-
dimensional search, referred to as a line search. 9
wylcd WY L CD 19
Line searches can be either exact or inexact. In- • β = (1−θ k )β +θ k β Hammel ;
k k k
exact line searches are generally preferred because
20
performing an exact line search at each iteration is • β hHZDY = (1 − θ k )β HZ + θ k β DY Hallal ;
k k k
computationally expensive in terms of both time
and memory 10,11 . In our research, we employ the • β HMLSFR = (1 − θ k )β MLS + θ k β FR
k k k
21
strong Wolfe line search, defined by the following Guefassa ;
conditions 12,13 :
• β hPRPHZ = (1 − θ k )β PRP + θ k β HZ
k 22 k k
T
f(x k + α k d k ) − f(x k ) ≤ δα k g d k , (3) Delladji ;
k
23
k+1 k | ≤ −σg d k .
|g T d T (4) • β k hFRBA = (1−θ k )β FR +θ k β k BA Delladji ;
k
k
hyb HS FR 24
• β = (1 − θ k )β + θ k β Djordjevi´c.
where 0 < δ ≤ σ < 1. And d k is the search direc- k k k
tion generated by the following rule:
From the literature, the AL-BAYATI and AL-
( 25
−g 0 , if k = 0 ASSADY (BA) method often performs better in
d k = (5) 26
−g k + β k−1 d k−1 , if k ≥ 0, practice than the Hager and Zhan (HZ) method,
while the HZ method exhibits stronger conver-
gence properties than the BA method. To com-
where g k is the gradient of f at the point x k
and β k is known as the CG parameter. Different bine the advantages of both the BA and HZ meth-
choices for the parameter β k correspond to vari- ods and develop a more efficient and robust algo-
ous conjugate gradient methods. Bellow are some rithm, we propose a new hybrid CG method based
well-known formulas: on a convex combination of these two methods
for solving unconstrained optimization problems
• β FR = ∥g k+1 ∥ 2 Fletcher and Reeves (FR); under suitable conditions. The corresponding pa-
k ∥g k ∥ 2 rameters are defined as follows:

14
• β DY = ∥g k+1 ∥ 2 Dai-Yuan method (DY) ;
k T
y d k
k HZ 1 ∥y k ∥ 2 T
g T y β k = T (y k − 2d k T ) g k+1 . (6)
k+1 k
HS
• β = Hestenes and Stiefel (HS); d y k d y k
k T k k
y d k
k 2
g T y β BA = ∥y k ∥ , (7)
• β LS = k+1 k Liu and Storey (LS). 15 k T
k T d y k
−g d k k
k
These methods are identical if f is a strongly where y k = g k+1 − g k and ∥.∥ stands for the Eu-
convex quadratic function and α k is obtained clidean norm.
by exact line search, as the parameters β k of This paper is organized as follows: In Section 2,
these methods will be equal, generating the we present the new hybrid conjugate gradient al-
same sequence {x k } k=0 . However, when ap- gorithm, detail its corresponding algorithm, and
∞
plied to a general nonlinear function with in- derive the parameter θ k using specific techniques.
exact line searches, these methods generate Furthermore, we establish the sufficient descent
different sequences {x k } k=0 , implying different condition under the strong Wolfe line search. In
∞
methods. 16–18 Section 3, we analyze the convergence proper-
The most important characteristic that combines ties of the proposed method. In Section 4, we
the methods FR and DY methods is their strong demonstrate the efficiency of our method through
26