Page 34 - IJOCTA-15-1

S. Ben Hanachi, B. Sellami, M. Belloufi / IJOCTA, Vol.15, No.1, pp.25-34 (2025)
Next, considering the above relation again and Finally: 30 if 0 < θ k < 1, then there exist two real
adding and subtracting the value θ k g k+1 , we get: numbers µ 1 , µ 2 such that 0 < µ 1 ≤ θ k ≤ µ 2 < 1,
and we have:
∥y k ∥ 2
hyb
d = −(1 − θ k + θ k )g k+1 + (1 − θ k ) d k
k+1 T
d y k
k T hyb T BA T HZ
d
d
d
1 T ∥y k ∥ 2 T g k+1 k+1 = (1 − θ k )g k+1 k+1 + θ k g k+1 k+1 ,
+θ k (y g k+1 − 2 d g k+1 )d k ,
T k T k
d y k d y k
k k
∥y k ∥ 2 ≤ (1 − µ 2 )g T d BA + µ 1 g T d HZ ,
= (1 − θ k )(−g k+1 + d k ) k+1 k+1 k+1 k+1
T
d y k
k
1
+θ k (−g k+1 + 29
T In, they proved that:
d y k
k
∥y k ∥ 2
T
2
T
(y g k+1 − 2 T d g k+1 )d k ). g T d BA ≤ −(1 − ϵ 1 )∥g k+1 ∥ , (14)
k
k+1 k+1
k
d y k
k
Finally, from the above relation, we obtain:
where, 0 < ϵ 1 << 1, from the above relations (13)
and (14), it follows that:
hyb BA HZ
d = (1 − θ k )d + θ k d .
k+1 k+1 k+1
□
7
2
hyb
2
2.1. The sufficient descent condition g T d ≤ −(1 − µ 2 )(1 − ϵ 1 )∥g k+1 ∥ − µ 1 ∥g k+1 ∥ .
k+1 k+1
8
In the following we prove that the search direction Therefore,
d k obtained by the new hybrid conjugate gradi-
2
ent satisfies the sufficient descent condition, which g T d hyb ≤ −K∥g k+1 ∥ , (15)
k+1 k+1
plays a crucial role in ensuring global convergence.
Based on the relation (12), we distinguish three where K = (1 − µ 2 )(1 − ϵ 1 ) + µ 1 7 8 .
cases:
Firstly: If θ k = 0, then Algorithm 1. (CCH, NDH)
n
hyb BA Initialization. Choose a starting point x 0 ∈ R ,
d = d ,
k+1 k+1 ϵ > 0, and 0 < δ ≤ σ < 1. Compute f(x 0 ) and
g 0 = ∇f(x 0 ). Consider d 0 = −g 0 , the initial
Therefore, the sufficient descent condition holds guess α 0 = 0, and k = 0.
for the hybrid method if it holds for the BA Step 1: If ∥g k ∥ < ϵ, then stop; otherwise, con-
method. Delladji, S., Belloufi, and Sellami prove tinue with Step 2.
in 29 that d BA satisfies the sufficient descent con- Step 2: Compute α k by the strong Wolfe line
k+1
dition for all k, under the strong Wolfe line search search (3), (4).
conditions. Step 3: Generate x k+1 = x k + α k d k .
Compute f(x k+1 ), g k+1 = ∇f(x k+1 ), and y k =
g k+1 − g k .
Secondly: If θ k = 1, then
T
Step 4: Compute the θ k parameter. If y g k −
k
hyb HZ ∥y k ∥ 2 T
d = d , 2 T d g k+1 = 0, then set θ k = 0. Otherwise,
k
k+1 k+1
d y k
k
compute θ k as follows:
Therefore, the sufficient descent condition holds
CCH algorithm (using the Conjugacy Condition):
for our hybrid method if it holds for the HZ
method, the following theorem (2), established by T
k
William W. Hager and Hong Zhang in, 26 proves θ k = g y k 2 .
∥y k ∥
T
T
the sufficient descent for the HZ method: y g k − 2 T d g k+1
k
k
d y k
k
T
Theorem 2. If d y k ̸= 0, and NDH algorithm (using the Newton Direction):
k
d k+1 = −g k+1 + τd k ,
T 2 T T
(y g k+1 − ∥y k ∥ − d g k+1 )d y k
k
k
k
d 0 = −g 0 , ∀τ ∈ [β k HZ , max{0, β k HZ }]. θ k = T T 2 T T .
(y g k+1 )(d y k ) − ∥y k ∥ (2d g k+1 + d y k )
k k k k
Then,
7 2
HZ
T
d
g k+1 k+1 ≤ − ∥g k+1 ∥ . (13) Step 5. If 0 < θ k < 1, compute β hyb as in
8 k
28