Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 5s (2024) 

 

372 
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A New Modified Secant Condition for Non-linear Conjugate Gradient 

Methods with Global Convergence 

 

Farhan Khalaf Muord1, Muna M. M. Ali2 

1,2 Department of Mathematics, College of Computer Science and Mathematics, Mosul University, Mosul, Iraq. 

1munamoh74@uomosul.edu.iq 

 

Article History: 

Received: 16-05-2024 

Revised: 23-06-2024 

Accepted: 11-07-2024 

Abstract:  

The Conjugate Gradient Methods(CGM) are well-recognized techniques for handling 

nonlinear optimization problems. Dai and Liao (2001) employ the secant condition 

approach, this study utilizes the modified secant condition proposed by Yabe-Takano 

(2004) and Zhang and Xu (2001), which is satisfied at each iteration through the 

implementation of the strong Wolf-line search condition. Additionally, please provide 

three novel categories of conjugate gradient algorithms of this nature. We examined 15 

well-known test functions. This novel approach utilises the existing gradient and 

function value to accurately approximate the goal function with high-order precision. 

The worldwide convergence of our novel algorithms is demonstrated under certain 

conditions. Numerical results are provided, and the efficiency is proven by comparing it 

to other approaches. 

Keywords: Conjugate Gradient technique, Un-constrained optimization, numerical 

studies, preconditioning, Sufficient descent condition, Convergence 

 

1. Introduction 

The unconstrained Problem for an optimization defended by: 

Min 𝑓〈𝑥〉          𝑥 ∈ 𝑅𝑛 ,                                                                                

(1) 

We have 𝑓: 𝑅𝑛 → 𝑅 is smooth and it’s ∇𝑓 is available Conjugate gradient(𝐶𝐺) method For Solving 

(1)is Iterative methods of the from 

    𝑥𝑘+1 = 𝑥𝑘 + 𝛼𝑘𝑑𝑘                                                                                             (2) 

Where 𝛼𝑘 > 0 is step size and 𝑑𝑘 is search direction, the 𝑑𝑘  is recursively known as: 

𝑑𝑘 = {
−𝑔𝑘                               𝑓𝑜𝑟  𝑘 = 1,
−𝑔𝑘 + 𝛽𝑘𝑑𝑘−1            𝑓𝑜𝑟  𝑘 ≥ 2,

                                                             (3) 

  𝑔𝑘 means ∇𝑓〈𝑥〉 and 𝛽𝑘 is a parameter, if 𝑓〈𝑥〉 is a strictly convex quadratic function  and 𝛼𝑘 is an 

exact one-dimension min .(1)-(3) is knows (CG)  method, Also, (1)-(2) is knows the nonlinear (CG) 

method. There are different general  unconstrained optimization problems, Famous prescription for 𝛽 

are the (LS) [1]. (PR) [2] and (HS) [3] which are as∶ 

βk
LS =

gk+1
T yk

dk
Tgk

                 (4) 

βk
PR =

gk    
T yk−1

‖gk−1‖2
                                                                                                      (5) 



Communications on Applied Nonlinear Analysis 

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βk
HS =

gk
Tyk−1

dK−1
T yk−1

                                                                             (6) 

 To prove the convergence of this approach, it is often needed that the step-size α_k satisfies the 

strong Wolfe condition, where 𝑦𝑘−1 is defined as 𝑔𝑘 minus 𝑔𝑘−1, and ‖. ‖ represents the Euclidean 

norm. 

f(xk) − f(xk + αkdk) ≥ δαkgk
Tdk                [4]                                                                   (7) 

      |g(xk + αkdk)dk|T ≤ −σdk
Tdk                   [4]                                                        (8) 

Where 0 < δ < 0.5 < σ < 1                

  Each method comes with its own advantages and disadvantages. see[5], [3] The Polak-Ribiere and 

Hestenes-Stiefel (HS) methods have comparable theoretical properties. Both of these methods are 

favored over the Liu-Storey (LS) method in terms of numerical performance. This is because they 

both restart after encountering a bad direction. However, the Yabe-Takano (YT) method, derived by 

Zhang et al in 2004, stands out .[6] and Zhang[7]and Xu [8] proposed  by Yabe-Takano (YT)[9]  we 

propose a three news formulas for 𝛽𝑘
𝑁1and βk

N2and βk
N3 by exploiting the modified secant condition 

in this paper is organized as follows ,in section2 we state a conjugacy condition and the formulas In 

section3,the modified secant condition is described In Section 4, We suggest a fresh requirement for 

conjugacy and develop novel equations for β.  In Section 5, we demonstrate the worldwide 

convergence of the latest conjugate gradient techniques under specific assumptions. Section 6 

includes the presentation of a few numerical trials. 

1.2 Yabe-Takano (YT) 

   Conjugate gradient algorithm generates a direction search such that the conjugacy condition holds, 

as, 

     di
TQdj = 0,                       ∀i ≠ j                                         (9)   

The matrix Q is positive definite for the quadratic objective function. For general nonlinear 

functions, the mean value theorem (M.V.T.) guarantees the existence of a value τ in the interval 

τ∈(0,1) such that 

  dk 
T yk−1 = αk−1𝑑𝑘

𝑇∇2f(xk−1 + ταk−1dk−1) 

Hence, it is acceptable to substitute (9) with the subsequent condition: 

      dk
T𝑦𝑘−1 = 0                                                                                            (10) 

Recently, extension of the CG has been studied Yabe-Takano. [10] using the secant condition of 

Quasi-Newton (QN) methods, 

    Hkyk−1 = Sk−1.                                                                                       (11) 

where   Hk is an inverse approximate for the Hessian and 𝑆𝑘−1 = 𝑥𝑘 − 𝑥𝑘−1. For quasi_Newton 

methods, the search direction 𝑑𝑘 can be calculated by:  

     dk = −Hkgk                                                                                           (12) 



Communications on Applied Nonlinear Analysis 

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By (11) and (12). We have that 

dk
Tyk−1 = −(Hkgk)Tyk−1 = −gk

T(Hkyk−1) = −gk
Tsk−1                   (13) 

By this relation, Dai and Liao replaced the conjugacy condition by the condition 

   𝑑𝑘
𝑇𝑦𝑘−1 = −𝑡𝑔𝑘

𝑇𝑠𝑘−1                                                                     (14) 

Where 𝑡 ≥  0 is a scalar. In the scenario where t=0, equation (11) simplifies to the standard 

conjugacy condition (10). Conversely, when t=1, equation (11) becomes equivalent to (10). In order 

to guarantee that the search direction dk meets this requirement, we can substitute equation (3) into 

(14) to obtain: 

  −gk
Tyk−1 + βkdk−1

T yk−1 = −tgk
Tsk−1    [10]                                               (15) 

2.New formulas for 𝛃𝐤
𝐍𝟏, 𝛃𝐤

𝐍𝟐, 𝛃𝐤
𝐍𝟑 

  We are developed a new conjugate gradient (CG) method in this part based on the work of Yabe-

Takano [10]. To achieve this work, we apply a modified secant condition (3.2) instead of the usual 

one (6).  Let 𝑧𝑘−1 be defined with a scalar parameter ρ ≥ 0: 

   {
𝑧𝑘−1 = 𝑦𝑘−1 + ρ (

𝜃𝑘−1

𝑠𝑘
𝑇𝑢𝑘−1

)                                  

θk−1 = 6(fk−1 − fk) + 3(gk−1 + gk)Tsk−1

            [11],[12]                                                (16) 

Building upon the same reasoning as in part 2, we examine the adjusted secant condition involving 

with  zk−1 

  Hkzk−1 = sk−1                                                                                                                                 (17) 

when 𝜌 = 0 and 𝜌 = 1,[6]   This situation aligns with the typical secant condition (6) and the revised 

secant condition (3.2) individually. It follows from (7) and (12) that: 

𝑑𝑘
𝑇𝑧𝑘−1 = −(𝐻𝑘𝑔𝑘)𝑇𝑧𝑘−1 = −𝑔𝑘

𝑇(𝐻𝑘𝑧𝑘−1) = −𝑔𝑘
𝑇𝑠𝑘−1                                                                 (18) 

  Considering this relationship, we substitute the conjugacy requirement with the updated condition.: 

𝑑𝑘
𝑇𝑧𝑘−1 = −𝑡𝑔𝑘

𝑇𝑠𝑘−1 ,                                                                                         (19)   

𝑑𝑘 = −𝑔𝑘 + 𝛽𝑘𝑑𝑘−1   

𝐻𝐾𝑧𝑘−1 = 𝑠𝑘−1 

𝐻𝐾𝑦𝑘−1 = 𝑠𝑘−1                                                       (20) 

𝑑𝑘𝑧𝑘 = −𝑔𝑘𝑠𝑘−1 

𝑑𝑘𝑧𝑘 = −𝑡𝑔𝑘𝑠𝑘−1                                                                                          

While   𝑧𝑘 = 𝑦𝑘 +
𝜃𝑘

‖𝑠𝑘‖2 𝑠𝑘                                                        (21) 

𝜃𝑘 = 6(𝑓𝑘 − 𝑓𝑘−1) + 3(𝑔𝑘 − 𝑔𝑘−1)𝑠𝑘 for 𝑡 ≥ 0                                                          (22) 

𝑑𝑘 = −𝐻𝑘𝑔𝑘                                                          (23) 



Communications on Applied Nonlinear Analysis 

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𝑑𝑘
𝑇𝑦𝑘 = 0 (Perry), this condition is true for all liner functions 

𝑑𝑘𝑦𝑘 = −𝐻𝑘𝑔𝑘𝑦𝑘                                                        (24) 

𝑑𝑘
𝑇𝑦𝑘 = −𝑔𝑘−1

𝑇 𝑠𝑘−1, this condition is true for all nonlinear functions 

𝑑𝑘𝑦𝑘 = −𝑡𝑔𝑘𝑠𝑘−1                                                                     (25) 

Multiply the equation(25)by 𝑧𝐾We get 

𝑑𝑘𝑧𝑘 = −𝑔𝑘𝑧𝑘 + 𝛽𝑘𝑑𝑘−1𝑧𝑘                                                         (26)     

−𝑡𝑔𝑘
𝑇𝑠𝑘 = −𝑔𝑘

𝑇𝑧𝑘 + 𝛽𝑘𝑑𝑘−1
𝑇 𝑧𝑘                                                      (27) 

−𝑡𝑔𝑘
𝑇𝑠𝑘 = −𝑔𝑘

𝑇𝑧𝑘 + 𝛽𝑘𝑑𝑘−1
𝑇 𝑦𝑘                                                         (28) 

(𝑔𝑘
𝑇𝑧𝑘 − 𝑡𝑔𝑘

𝑇𝑠𝑘)= 𝛽𝑘𝑑𝑘
𝑇𝑦𝑘−1                                                                       (29) 

𝛽𝑘
𝑁1 =

𝑔𝑘
𝑇(𝑧𝑘−𝑡𝑠𝑘)

𝑑𝑘
𝑇𝑦𝑘−1

                                  (30) 

In the next section ,we proved the global convergence of the new methods (30) , following the Lio-

story methods , we are get by using (30)  

𝛽𝑘
𝑁1 = 𝑀𝑎𝑥 {

𝑔𝑘
𝑇𝑧𝑘 

𝑑𝑘−1
𝑇 𝑦𝑘−1

, 0} − 𝑡 {
𝑔𝑘

𝑇 𝑠𝑘 

𝑑𝑘 
𝑇 𝑦𝑘−1

, 0}                                                                                         (31) 

 this case, if (ELS) then we have 𝜃𝑘−1 = 0, 𝑧𝑘−1 = 𝑦𝑘−1 the new parameters become𝑔𝑘
𝑇𝑦𝑘−1/𝑑𝑘

𝑇𝑦𝑘−1 

. Thus, our equation (30) simplifies to the Hestenes-Stiefel formula when considering linear 

conjugate gradient techniques.  . 

Similarly we have derive  , βk
N2, βk

N3 

βk
N2 =

𝑔𝑘
𝑇(𝑧𝑘−𝑡𝑠𝑘)

‖𝑔𝑘−1‖2                                                           (32) 

where 𝑧𝑘 = 𝑦𝑘 +
1

3

𝜃𝑘

‖𝑠𝑘‖2 𝑠𝑘     [13]                                                               (33) 

and 

𝜃𝑘 = 2(𝑓𝑘 − 𝑓𝑘+1) + (𝑔𝑘+1 + 𝑔𝑘)𝑇𝑠𝑘                                                    (34) 

and 

βk
N3 =

gk
T(zk−tsk)

−gk−1
T dk−1

                                                          (35) 

 where 𝑧𝑘 = 𝑦𝑘 +
2

3
 

𝜃𝑘

‖𝑠𝑘‖2
𝑠𝑘 [14]                                                               (36) 

𝜃𝑘 = 4(𝑓𝑘−1 − 𝑓𝑘) + 2(𝑔𝑘 + 𝑔𝑘+1)𝑇𝑠𝑘−1  [14][15]                                              (37)   

2.1 Algorithm 

   Step1:Take  𝑥0 ∈ 𝑅𝑛 ,and 0 < 𝛿 ≤ 𝜎 < 1 ,Calculate 𝑓(𝑥0) and 𝑔0 = ∇𝑓(𝑥0) , set 

 𝑑0 = −𝑔0  for 𝑘 = 0  



Communications on Applied Nonlinear Analysis 

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Step2: Compute 𝛼𝑘 Satisfying Strong "Wolfe Condition ")3.10) (3.11) and then Compute 

 𝑥𝑘+1 = 𝑥𝑘 + 𝛼𝑘𝑑𝑘 

Step3:If(‖𝑔𝑘‖∞ ≤ 10−10 or (|𝛼𝑘𝑔𝑘𝑑𝑘| ≤ 10−10|𝑓𝑘|) is Satisfy then stop. 

Step4: Compute the new search direction : 

                      𝑑𝑘 = −𝑔𝑘 + 𝛽𝑘𝑑𝑘−1 

  If the restart criterion of Powell, such that |gk
T𝑔𝑘−1| ≥ 0.2‖𝑔𝑘‖2 is Satisfied, then set 𝑑𝑘 = −𝑔𝑘; 

otherwise, define  

Step5:Compute the new Parameters βk
N1, βk

N2, βk
N3from(30,31,34)respectively 

Step6: Set 𝐾 = 𝐾 + 1 and go to Step2. 

3.  Convergence analysis 

3.1 Introductory   

      In this section we are position verify global convergence of the new methods . 

3.1 Hypothesis 1: see [10].  

Dia et al. demonstrated that any CG algorithm utilizing the powerful Wolf line search yields the 

subsequent beneficial outcome [16],[17].   

3.2 Lemma 

From he hypothesis 1 holds .we have for any  CG method in the form (2)-(3), where 𝑑𝑘satisfies the 

D.C. in (26) ,where  𝛼𝑘 is gets by the strong Wolf line search (28)-(29). Let 𝜑 ∈ [0,4] be given[18]. 

If  

∑
‖𝑔𝑘‖𝜑

‖𝑔𝑘‖2
= ∞

𝑘≥1

 

Then the following holds 

lim
𝑘→∞

‖𝑔𝑘‖ = 0 

by then. . . if 𝜃 = 0 ⇒ lim𝑘→∞  ∥∥𝑔𝑘∥∥ = 0 

3.3 Theorem 

Assuming assumption 1 is valid and f is a uniformly convex function, let's take a look at the 

conjugate gradient method using equation (15). In this method, the values of dk and 𝑢𝑘 must satisfy 

the descent condition (26) and condition (5.9) respectively, while αk is determined through the strong 

Wolfe line search algorithm. If 𝐿 = 𝜇 then our method with 𝜌 ≥ 0 satisfies lim
𝑘→∞

‖𝑔𝑘‖ = 0. If L > 𝜇, 

then our method 0 ≤ 𝜌 <
𝐿

3(𝐿−𝜇)
 satisfies  lim

𝑘→∞
‖𝑔𝑘‖ = 0. 

 

 



Communications on Applied Nonlinear Analysis 

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Proof: from paper [10]we have that  

From {
𝑧𝑘−1 = 𝑦𝑘−1 + ρ (

𝜃𝑘−1

𝑠𝑘
𝑇𝑢𝑘−1

)                                  

𝜃𝑘−1 = 6(𝑓𝑘−1 − 𝑓𝑘) + 3(gk−1 + gk)𝑇𝑠𝑘−1

    

 and 𝑓𝑘−1 − 𝑓𝑘 ≥ 𝑔𝑘
𝑇𝑠𝑘−1 +

𝜇

2
 ‖𝑠𝑘−1‖2  

and  𝜇‖𝑠𝑘−1‖2 ≤ 𝑠𝑘−1
𝑇 𝑦𝑘−1 ≤ 𝐿‖𝑠𝑘−1‖2                     

   𝑠𝑘−1
𝑇 𝑧𝑘−1 = 𝑠𝑘−1

𝑇 𝑦𝑘−1 + 𝜌𝜃𝑘−1 

   = 𝑠𝑘−1
𝑇 𝑦𝑘−1 + 6𝜌(𝑓𝑘−1 − 𝑓𝑘) + 3𝜌(𝑔𝑘−1 + 𝑔𝑘)𝑇𝑠𝑘−1 

  𝑧𝑘−1 = 𝑦𝑘−1 + 𝜌 (
𝜃𝑘−1

𝑠𝑘−1
𝜋 𝑢𝑘−1

𝑢𝑘−1) . 

 𝜃𝑘−1 = 6(𝑓𝑘−1 − 𝑓𝑘) + 3(𝑔𝑘−1 + 𝑔𝑘)⊤𝑠𝑘−1 

   𝐻𝑘𝑧𝑘−1 = 𝑠𝑘−1 

 𝑠𝑘−1
⊤ 𝑧𝑘−1 = 𝑠𝑘−1

⊤ 𝑦𝑘−1 + 𝜌𝜃𝑘−1 

  = 𝑠𝑘−1
𝑇 𝑦𝑘−1 + 6𝜌(𝑓𝑘−1 − 𝑓𝑘) + 3𝜌(𝑔𝑘−1 + 𝑔𝑘)𝑇𝑠𝑘−1 

  ≥𝑠𝑘−1
⊤ 𝑦𝑘−1 + 6𝜌 (−𝑔𝑘

⊤𝑠𝑘−1 +
𝜇

2
∥∥𝑠𝑘−1∥∥2) + 𝜌3(𝑔𝑘−1 + 𝑔𝑘)𝑇𝑠𝑘−1 

 = 𝑠𝑘−1
𝑇 𝑦𝑘−1 − 3𝜌𝑔𝑘

𝑇𝑠𝑘−1 + 3𝜌𝑔𝑘−1
𝑇 𝑠𝑘−1 + 3𝜌𝜇∥∥𝑠𝑘−1∥∥2

 

  = 𝑠𝑘−1
𝑇 𝑦𝑘−1 − 3𝜌𝑠𝑘−1

𝑇 𝑦𝑘−1 + 3𝜌𝜇∥∥𝑠𝑘−1∥∥2
 

  = (1 − 3𝜌)𝑠𝑘−1
𝑇 𝑦𝑘−1 + 3𝜌𝜇∥∥𝑠𝑘−1∥∥2

 

   ≥ (1 − 3𝜌)𝑠𝑘−1
𝑇 𝑦𝑘−1 +

3𝜌𝜇

𝐿
𝑠𝑘−1

𝑇 𝑦𝑘−1 

   =
𝐿−3𝜌(𝐿−𝜇)

𝐿
𝑆𝑘−1

𝑇 𝑦𝑘−1                                                                                   (38) 

We have 0 ≤ 𝜌 <
𝐿

3(𝐿−𝜇)
, and we have: 

𝑠𝑘−1
𝑇 𝑧𝑘−1 ≥ {

𝐿 − 3𝜌(𝐿 − 𝜇

𝐿
}  𝜇‖𝑠𝑘−1‖2 

∥∥𝑑𝑘∥∥ = ∥∥−𝑔𝑘 + 𝛽𝑘𝑑𝑘−1∥∥ 

=
∥∥
∥∥−𝑔𝑘 +

𝑔𝑘
⊤(𝑦𝑘 − 𝑡𝑠𝑘−1)

𝑑𝑘−1
⊤ 𝑦𝑘−1

𝑑𝑘−1
∥∥
∥∥ 

≤ ∥∥𝑔𝑘∥∥ +
∥∥𝑔𝑘∥∥(∥∥𝑦𝑘−1∥∥ + 𝑡∥∥𝑠𝑘−1∥∥

|𝑑𝑘−1
𝜏 𝑦𝑘−1|

∥∥𝑑𝑘−1∥∥ 

≤ ∥∥𝑔𝑘∥∥ +
∥∥𝑔𝑘∥∥(𝐹∥∥𝑠𝑘−1∥∥ + 𝑡∥∥𝑠𝑘−1∥∥

|𝑠𝑘−1
𝑇 𝑧𝑘−1|

∥∥𝑠𝑘−1∥∥ 



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≤ ∥∥𝑔𝑘∥∥ +
(𝐹 + t)∥∥𝑔𝑘∥∥ ∥∥𝑠𝑘−1∥∥2

𝑓1∥∥𝑠𝑘−1∥∥2  

= (1 +
𝐹 + 𝑡

𝑓1
) ∥∥𝑔𝑘∥∥ 

≤
(𝑓1 + 𝐹 + 𝑡)�̅�

𝑓1
 

∑  
1

∥ 𝑑1𝑑|2
≥ {

𝑓1

(𝑓1 + 𝐹 + 𝑡)𝛾
}

2

∑  

𝑘≥1

 1 = ∞ 

By Theorem 3. If 𝜎 = 0 ⇒ lim
𝑘→∞

‖𝑔𝑘‖ = 0 

4. Numerical results 

  This section provides the numerical results that assessed the effectiveness of the conjugate gradient 

algorithms employing the Fletcher-Reeves (FR), Polak-Ribiere (PR), and Liu-Storey (LS) 

techniques.   This information is also available in reference [19].   

The program's requirements for stopping ‖𝑔𝑘+1‖ ≤ 10−5 and written in (FORTRAN90). Table 1 

shows the number of the function (NOF) and the number of the iteration (NOI) and confirms that the 

new methods are superior (NI) with dimension n=1000, 10000[19],[20] 

Table (1) Comparison between the new 𝛽𝑁1 , 𝛽𝑁2, 𝛽𝑁3 methods against 𝛽𝐻𝑆 , 𝛽𝑃𝑅 , 𝛽𝐿𝑆 methods 

for the Total of 30-problems with n=1000 , 10000 

𝛽𝑁1 𝛽𝑁2 𝛽𝑁3 βHS 𝛽𝑃𝑅 𝛽𝐿𝑆 N P.No. 

Fns NOF NOI NOF NOI NOF NOI NOF NOI NOF NOI NOF NOI 

71 23 76 26 77 27 103 52 103 53 102 52 1000 1 

80 29 82 32 80 29 124 49 127 52 132 51 10000 

70 13 80 19 72 15 91 32 112 50 92 21 1000 2 

59 20 61 28 59 21 73 32 73 42 71 36 10000 

67 21 73 23 68 21 81 37 91 37 82 36 1000 3 

98 35 106 43 101 35 122 47 122 61 113 47 10000 

60 38 70 45 55 31 83 63 93 65 71 51 1000 4 

69 44 64 39 51 30 93 72 89 53 72 53 10000 

38 18 47 23 41 20 51 39 60 36 54 33 1000 5 

45 20 54 28 49 21 60 34 66 40 61 23 10000 

30 14 30 14 29 15 47 32 47 32 45 27 1000 6 

32 16 32 16 30 15 52 41 52 37 51 41 10000 

102 13 115 15 115 14 153 37 161 43 142 41 1000 7 

90 12 109 14 95 14 104 26 121 28 109 26 10000 

90 11 103 15 99 12 102 23 115 27 111 24 1000 8 

166 15 188 19 178 16 182 29 200 31 192 28 10000 

60 19 129 17 120 15 82 37 152 42 142 41 1000 9 

195 18 200 19 199 19 226 42 302 52 302 52 10000 

95 15 121 17 99 17 107 27 133 29 111 29 1000 10 

163 14 198 21 185 18 182 30 233 47 162 41 10000 

105 32 112 39 108 35 143 52 209 62 142 51 1000 11 



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133 40 145 45 139 45 145 61 182 61 172 61 10000 

55 23 56 25 55 24 82 51 82 52 82 53 1000 12 

50 20 60 33 51 23 82 44 83 51 83 41 10000 

52 25 66 30 60 25 64 37 78 42 72 37 1000 13 

160 31 191 41 170 38 173 43 203 54 184 50 10000 

80 17 90 35 83 18 92 29 122 52 111 41 1000 14 

70 12 77 20 77 20 102 34 103 41 103 41 10000 

80 17 90 35 83 18 83 31 92 41 101 30 1000 15 

100 58 106 70 102 60 122 70 126 83 120 81 10000 

94 11 100 14 98 13 106 25 112 26 110 27 1000 16 

94 16 109 14 109 19 106 30 121 26 122 31 10000  

2753 710 3140 874 2937 743 3416 1288 3965 1448 3720 1317  Total 

 

Clearly, we have from the Table (2) that New1algorithm beats (HS) algorithm in about (44%) NOI; 

(21%) NOF, also, we have the New2 algorithm beats (PR) algorithm in about (40%) NOI,(21%) 

NOF then we have the New3 algorithm beats (LS) algorithm in about (45%) NOI,(19%) NOF. 

Table2: percentage modified of the new algorthims 

 HS algorithm 𝛽𝑁1 PR algorithm 𝛽𝑁2 LS algorithm 𝛽𝑁3 

NI 100% 56% 100% 60% 100% 55% 

NF 100% 79% 100% 79% 100% 81% 

 

The charts below (Fig(1), Fig(2)( show the comparison of the new algorithm with similar algorithms 

(HS, PR, LS) based on the number of iterations and the number of function calculations, 

respectively. 

We used More,Dolan to compare the new methods with the classical methods, based on the number 

of iterations and the number of function calculations. 

 
Figure 1:  Number of iteration comparing 

bewteen New methods to standred methods 

 

Figure 2:  Number of function evaluation 

comparing bewteen New methods to standred 

method 

 



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5. Conclusion 

  1. The CG Methods are proposed for solving nonlinear Optimization  Problems 

  2.   Adequate decrease and worldwide convergence can be achieved under certain conditions. 

       The numerical findings shown in the previously mentioned figure. 

3.The new algorithms (𝛽𝑁1, 𝛽𝑁2, 𝛽𝑁3 )have prove its efficiency through results in table(1) and(2)      

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 Appendix 

The Test Function For Unconstrained Optimization 

No. The Test Function 

1 Beale 

2 Trigonametric 

3 Generalized Quadratic 

4 Hager 

5 Diagonal 1 

6 Diagonal 2 

7 EDENSCH  

8 EDENSCHNB  

9 FLETCHER  

10 NONDIA 

11 Extend Rosenbrock 

12 Extend Powell 

13 Extend Hiebert 

14 Extend Wood 

15 Extend Quadratic