EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5932
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A Globally Convergent Conjugate Gradient Method
Incorporating Perry’s Parameter for Unconstrained

Optimization

Hasan Hazim Jameel1, Alaa Luqman Ibrahim2,∗, Neven E. Zaya3

1 Department of Mathematics, College of Basic Education, University of Duhok,
Kurdistan Region-Iraq
2 Department of Mathematics, College of Science, University of Zakho,
Kurdistan Region-Iraq
3 Management Information System, Administrative Technical Institute,
Duhok Polytechnic University, Kurdistan Region-Iraq

Abstract. To develop a conjugate gradient method that is both theoretically robust and prac-
tically effective for solving unconstrained optimization problems, this paper introduces a novel
conjugate gradient method incorporating Perry’s parameter along with the gradient-difference
vector, as suggested by Powell, to enhance performance. The proposed method satisfies the de-
scent condition, and its global convergence is established under standard assumptions. To assess
its effectiveness, the method was tested on a diverse set of unconstrained optimization problems
and compared against well-known conjugate gradient methods. Numerical experiments indicate
that the proposed method outperforms classical approaches in terms of iteration count, function
evaluations, and computational time. The results confirm the robustness and efficiency of the
proposed method, making it a competitive choice for large-scale optimization problems.

2020 Mathematics Subject Classifications: 49-XX, 93-XX, 90Bxx, 90-XX.

Key Words and Phrases: Unconstrained Optimization, Conjugate Gradient Methods, Global
Convergence, Numerical Performance.

1. Introduction

Optimization involves the minimization or maximization of an objective function. A
key subset of optimization is unconstrained optimization, which focuses on minimizing
a function of real variables without constraints. The general unconstrained optimization
problem can be formulated as:

min{f(x), x ∈ Rn}, (1)

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5932

Email addresses: hasan.jameel@uod.ac (H. H. Jameel),
alaa.ibrahim@uoz.edu.krd (A. L. Ibrahim), neven.zaya@dpu.edu.krd (N. E. Zaya)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 2 of 15

where f is a smooth function with an available gradient [1]. Several numerical methods
have been developed to solve such problems, including the Steepest Descent (SD) method,
Newton’s method, Conjugate Gradient (CG) methods, and Quasi-Newton (QN) methods.
This paper focuses on CG methods due to their efficiency and scalability, particularly for
large-scale optimization problems. CG algorithms are widely applied in various fields. In
engineering, they are used for solving problems in structural design, fluid dynamics, and
control systems [2]. In data science and machine learning, CG methods play a crucial role
in optimizing loss functions for regression, classification, and neural network training [3–8].
Additionally, in signal and image processing, CG methods are used for tasks such as image
reconstruction, denoising, and signal recovery [9]. Scientific computing also heavily relies
on CG algorithms, particularly for solving large sparse linear systems in finite element
analysis and computational physics.

A key advantage of CG methods is their low memory requirement [10], which makes
them suitable for high-dimensional problems. Furthermore, they exhibit rapid conver-
gence for specific classes of functions. The foundation of CG methods was laid in 1952 by
Hestenes and Stiefel [11], who introduced the CG method for unconstrained linear opti-
mization. Later, in 1964, Fletcher and Reeves extended the CG method to solve nonlinear
unconstrained minimization problems [12].

In this paper, we propose a new conjugate gradient method that enhances computa-
tional efficiency while ensuring the satisfaction of the descent condition, and convergence
properties. Our approach introduces a novel CG parameter that optimizes search di-
rections and improves convergence rates, making it more effective in solving large-scale
optimization problems.
CG methods generate a sequence {xk} defined as:

xk+1 = xk + αkdk, (2)

where x0 is a randomly chosen initial starting point. The step size αk is determined using a
one-dimensional search known as a line search [13]. Line searches can be exact or inexact,
with the latter being preferred due to computational efficiency [14, 15]. In our research,
we employ the strong Wolfe line search, which satisfies the following conditions [16, 17]:

f(xk + αkdk) ≤ f(xk) + δαkg
T
k dk, (3)

g(xk + αkdk)
Tdk ≥ µgTk dk, (4)

where 0 < δ < µ < 1. The strong Wolfe-Powell (SWP) condition further includes:

|g(xk + αkdk)
Tdk| ≤ µ|gTk dk|. (5)

The search direction dk is determined as:

dk+1 =

{
−g1, k = 0,

−gk+1 + βkdk, k ≥ 1.
(6)



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 3 of 15

where gk = ∇f(xk) and βk is a CG parameter that varies depending on the CG method
used. The search direction must satisfy the descent condition:

gTk+1dk+1 ≤ 0. (7)

Various CG methods arise from different choices of βk, including Hestenes-Stiefel (HS)
[11], Fletcher-Reeves (FR) [12], Polak-Ribière-Polyak (PRP) [18, 19], Liu-Storey (LS)
[20], Dai-Yuan (DY) [21], and Conjugate-Descent (CD) [22]:

βHS
k =

gTk+1yk

yTk dk
, (8)

βFR
k =

∥gk+1∥2

∥gk∥2
, (9)

βPRP
k =

gTk+1yk

∥gk∥2
, (10)

βLS
k =

gTk+1yk

−gTk dk
, (11)

βDY
k =

∥gk+1∥2

yTk dk
, (12)

βCD
k =

∥gk+1∥2

−gTk dk
, (13)

where yk = gk+1 − gk and ∥ · ∥ denotes the Euclidean norm in Rn. In general, the DY
and FR methods with inexact line searches demonstrate good convergence performance,
though their computational efficiency is inferior to the PRP method. In recent years,
many scholars have proposed improved CG methods to balance numerical performance and
convergence properties [23].For more details on the various developments in CG methods,
see [24–27].

This paper is structured as follows: Section 2 presents the new conjugate gradient
algorithm along with its details. Section 3 establishes the descent condition. Section
4 analyzes the convergence properties of the proposed method. Section 5 demonstrates
its efficiency through numerical comparisons with standard methods using test problems
from the CUTE collection and unconstrained optimization problems. Finally, Section 6
concludes the paper.

2. Proposed Method and Algorithm

In this section, we introduce a newly proposed CG method for solving nonlinear opti-
mization problems. This method is based on the Perry parameter [28]:

βPerry
k =

gTk+1(yk − vk)

dTk yk
, (14)



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 4 of 15

and employs a gradient-difference vector as suggested by Powell [29]:

ȳk = yk + (1− θ)(Gksk − yk), (15)

where 0 < θ < 1.
Additionally, to incorporate second-order information without computing the Hessian

explicitly, Andrei [30] proposed a nonlinear CG algorithm where the Hessian-vector prod-
uct is approximated using finite differences:

y∗k = yk + (1− θ)
(yk
σ

− yk

)
, (16)

where σ =
2
√
ϵm(1+∥xk+1∥)

∥vk∥ , and ϵm is the machine precision. Now, let us define the updated
vectors as follows:

y∗k =
yk
σ

− θyk
σ

+ θyk, (17)

and for the yk vector:

y∗k =

(
1− θ

σ
+ θ

)
yk. (18)

By substituting y∗k into the Perry formula, we obtain the new expression for the con-
jugate gradient parameter:

βnew
k =

gTk+1(y
∗
k − vk)

dTk y
∗
k

=
gTk+1y

∗
k − gTk+1vk

dTk y
∗
k

. (19)

After performing some algebraic manipulations, we derive the final form of βnew
k :

βnew
k =

gTk+1yk

dTk yk
− σ

(1− θ + σθ)
·
gTk+1vk

dTk yk
, (20)

where σ =
2
√
ϵm(1+∥xk+1∥)

∥vk∥ , and 0 < θ < 1.
The newly developed conjugate gradient methods aim to improve the efficiency and

convergence characteristics of traditional CG approaches, particularly for high-dimensional
optimization problems.

Note: When θ = 1, the expression in equation (20) reduces to the traditional Perry
formula, meaning we revert to the original conjugate gradient method that does not use
finite differences or second-order approximations.



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 5 of 15

Algorithm 1 New Conjugate Gradient Method for Unconstrained Optimization

1: Initialization Choose an initial point x0 ∈ Rn, parameters 0 < δ < µ < 1, and
tolerance ϵ > 0. Set k = 0 and d1 = −g1.

2: Stopping Condition If ∥gk∥ ≤ ϵ, stop.
3: Compute Step Size Compute αk using the weak Wolfe conditions (Equations (3)

and (4)).
4: Update Update the point:

xk+1 = xk + αkdk. (21)

5: Compute Direction Compute:

dk+1 = −gk+1 + βkdk. (22)

Choose βk using Equation (20).
6: Repeat Set k = k + 1 and go to Step 2.

3. Descent Property and Sufficient Descent Condition

In this section, we establish the descent condition for the proposed algorithm.

Theorem 1. If the search direction dk+1 is generated by equation (6) with βNew
k from

equation (20), then the descent condition holds:

gTk+1dk+1 ≤ 0. (23)

Proof. Multiplying both sides of equation (6) by gTk+1 and substituting βNew
k from

equation (20), we obtain:

gTk+1dk+1 = −gTk+1gk+1 +
gTk+1yk

dTk yk
gTk+1dk − (

σ

(1− θ) + σθ
)
gTk+1vk

dTk yk
gTk+1dk. (24)

If the search direction is exact, then it satisfies the descent condition:

gTk+1dk+1 = −∥gk+1∥2 ≤ 0. (25)

For an inexact search direction where gTk+1dk ̸= 0, we note that the first two terms of
equation (24) satisfy:

−gTk+1gk+1 +
gTk+1yk

dTk yk
gTk+1dk ≤ 0, (26)

which follows from the HS method’s descent property from the inequality:

gTk+1yk ≤ ∥gk+1∥∥yk∥.

−∥gk+1∥2 +
gTk+1yk

dTk yk
dTk yk ≤ −∥gk+1∥2 + ∥gk+1∥∥yk∥.



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 6 of 15

gTk+1dk+1 ≤ −
(
1− ∥yk∥

∥gk+1∥

)
∥gk+1∥.

Since 0 ≤ ∥yk∥
∥gk+1∥ ≤ 1, we have:

−∥gk+1∥2 +
gTk+1yk

dTk yk
dTk yk ≤ 0.

It remains to prove that the third term of (24) is less than or equal to zero.

Since σ =
2
√
ϵm(1+∥xk+1∥)

∥vk∥ ≥ 0 and 0 < θ < 1. Thus, we conclude that:

σ

(1− θ) + σθ
≥ 0. (27)

Therefore, equation (24) can be rewritten as:

gTk+1dk+1 ≤ −(
σαk

(1− θ) + σθ
)
(gTk+1dk)

2

dTk yk
. (28)

Since αk, d
T
k yk, and (gTk+1dk)

2 are non-negative, we obtain:

gTk+1dk+1 ≤ 0. (29)

This completes the proof.

4. Convergence Condition for the New Method

This section discusses the convergence properties of the proposed algorithm. We in-
troduce the following mild assumptions in order to achieve global convergence.

Assumption 1:

• The level set S = {x | f(x) ≤ f(x0)} is bounded.

• In some neighborhood N of δ, f is continuously differentiable, and its gradient is
Lipschitz continuous with a Lipschitz constant L > 0. That is,

∥g(x)− g(y)∥ ≤ L∥x− y∥ ∀x, y ∈ S (30)

From these assumptions, it follows that there exists a positive constant b such that

∥g(x)∥ ≤ b ∀x ∈ S. (31)

Based on these assumptions, the global convergence of the new algorithm can be
established as follows:



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 7 of 15

Lemma 1. [31] Let assumptions 1 hold. Consider methods (1) and (2), where dk+1 is a
descent direction, and αk satisfies the standard Wolfe line search. If∑

k≥1

1

∥dk+1∥2
= ∞, (32)

then
lim
k→∞

inf ∥gk+1∥ = 0. (33)

Based on the previous discussion, we now establish the global convergence of the new
algorithm:

Theorem 2. If assumptions 1 is satisfied, and the sequences {xk}, {dk}, {gk}, {αk} are
generated by Algorithm 1, then it holds that

lim
k→∞

inf ∥gk+1∥ = 0. (34)

Proof. From equations (2) and (20), we have

∥dk+1∥ ≤ ∥gk+1∥+

∣∣∣∣∣gTk+1yk

dTk yk
− σ

(1− θ) + σθ

gTk+1vk

dTk yk

∣∣∣∣∣ ∥dk∥. (35)

Simplifying this further, we obtain

∥dk+1∥ ≤ ∥gk+1∥+

(∣∣∣∣∣gTk+1yk

dTk yk

∣∣∣∣∣+ σ

(1− θ) + σθ

∣∣∣∣∣gTk+1vk

dTk yk

∣∣∣∣∣
)
∥dk∥. (36)

Since gTk+1vk ≤ αkd
T
k yk, and using equation (28) along with the Lipschitz condition

∥yk∥ ≤ L∥vk∥, we also have gTk+1yk ≤ LgTk+1dk, where L > 0. Therefore,

∥dk+1∥ ≤ M +

(
L+

σαk

(1− θ) + σθ

)
∥dk∥. (37)

Since ∥vk∥ = ∥x− xk∥ and D = max{∥x− xk∥ | ∀x, xk ∈ R}, equation (37) becomes

∥dk+1∥ ≤ M +

(
L+

σαk

(1− θ) + σθ

)
D = ϕ. (38)

Thus, we have ∑
k≥1

1

∥dk+1∥2
≥
∑
k≥1

1

ϕ2
= ∞, (39)

which implies ∑
k≥1

1

∥dk+1∥2
= ∞. (40)

By Lemma 1, we conclude that

lim
k→∞

inf ∥gk∥ = 0, (41)

which completes the proof.



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 8 of 15

5. Numerical Experiments

In this section, we present the computational performance of a Matlab implementation
of the CG algorithm on a set of unconstrained optimization test problems from the CUTE
library [32, 33] and other sources of unconstrained problem sets [34]. These problems
represent a wide range of problem dimensions, from small-scale cases to large-scale cases.

This comparison is based on several performance metrics: the number of iterations
(NI), the total frequency of objective function evaluations (NF), and the CPU time
(CPUT) in seconds. The computational performance of the proposed methods is com-
pared with the HS and Perry methods. Each algorithm applied the strong Wolfe line
search conditions, with parameters set to δ = 0.01 and µ = 0.3. Iterations were termi-
nated if any of the following conditions were satisfied:

• ∥gk∥ < 10−6, where gk denotes the Euclidean norm.

• The number of iterations exceeded 2000.

• The computation time surpassed 500 seconds.

However, it is important to note that in some cases, the method failed to converge to a
solution. In such instances, the results were marked as NaN to indicate that the algorithm
could not produce a valid solution within the given constraints. All codes are written in
Matlab, and the computational experiments were performed on a personal laptop running
Windows 10, with an Intel(R) Core(TM) i7-6600U CPU @ 2.60GHz (2.81 GHz) and 8.00
GB of RAM.

Table 1 presents the results in terms of the NI, number of NF, and CPUT. Table 2
provides a detailed comparison of the proposed method with the classical HS and Perry
algorithms, highlighting the percentage improvements achieved by the new method.

A comprehensive analysis of these results was conducted using the performance profile
tool introduced by Dolan and Moré [35]. This tool provides a cumulative distribution
function ps(τ), enabling a unified and fair comparison of the algorithms across all metrics.
The performance profiles for the NI, NF, and CPUT are displayed in Figures 1a, 1b, and
1c, respectively.

Table 1: Numerical results of the proposed method compared with HS and Perry methods

Test Function N
HS Perry New New (ϵm = 1)

NI NF CPUT NI NF CPUT NI NF CPUT NI NF CPUT

’cosine’ 500 55 133 0.033755 31 85 0.009681 37 93 0.012033 31 83 0.016694

’cosine’ 1000 51 117 0.023636 26 79 0.01372 27 78 0.014678 26 79 0.014862

’cosine’ 5000 34 86 0.075612 31 97 0.08309 29 85 0.075538 31 98 0.080135

’cosine’ 10000 193 248 0.401999 20 71 0.130764 20 70 0.119581 20 71 0.13456

’dixmaana’ 1500 20 66 0.092721 14 58 0.084754 15 56 0.07047 14 58 0.078474

’dixmaana’ 3000 24 73 0.22983 18 66 0.21372 14 61 0.17963 18 66 0.218552



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 9 of 15

’dixmaana’ 15000 21 73 0.90865 19 71 0.827259 15 69 0.909347 19 71 0.938866

’dixmaana’ 30000 16 70 1.910883 16 69 1.713247 18 72 2.13092 16 69 1.629957

’dixmaanb’ 1500 10 53 0.090052 15 63 0.101415 16 64 0.07883 15 63 0.072512

’dixmaanb’ 3000 13 58 0.17841 19 66 0.192647 20 68 0.216327 19 66 0.187179

’dixmaanb’ 15000 14 65 0.844221 18 69 0.937515 19 69 0.903137 18 69 0.841963

’dixmaanb’ 30000 21 74 1.728917 18 70 1.732772 19 73 1.791905 18 70 1.777622

’dixmaanc’ 3000 23 72 0.215691 19 68 0.200968 13 61 0.183541 19 68 0.214485

’dixmaanc’ 15000 19 72 0.901367 19 72 0.887783 18 69 0.859604 19 72 0.918407

’dixmaane’ 1500 373 422 0.526614 186 298 0.327686 154 250 0.341556 158 266 0.314185

’dixmaane’ 3000 456 506 1.58176 208 361 1.002094 209 360 1.099917 231 378 1.140235

’dixmaane’ 15000 744 797 10.1206 343 554 6.977925 315 531 6.760724 430 682 8.448322

’dixmaane’ 24000 999 1054 20.09337 394 629 12.0246 445 760 14.85598 451 706 13.26736

’dixmaanf’ 15000 494 547 7.602266 277 469 5.851253 250 442 5.774908 428 568 7.05392

’dixmaanf’ 24000 1356 1418 28.5487 316 491 9.888144 254 429 8.428413 310 496 9.647825

’dixmaang’ 1500 NaN NaN NaN 167 271 0.382597 152 268 0.315178 153 245 0.301801

’dixmaang’ 3000 NaN NaN NaN 150 257 0.756952 203 327 0.945721 156 253 0.73545

’dixmaang’ 15000 373 427 5.115567 223 358 4.527224 265 424 5.191343 246 406 4.989654

’dixmaang’ 30000 1415 2595 62.24052 122 212 4.965631 168 296 7.095215 139 248 5.888025

’dixmaanh’ 1500 NaN NaN NaN 114 194 0.229408 133 244 0.36558 114 214 0.287352

’dixmaanh’ 3000 NaN NaN NaN 188 288 0.834066 159 279 0.814102 171 293 0.87006

’dixmaani’ 300 935 980 0.385063 407 638 0.207102 352 602 0.219785 402 642 0.211295

’dixmaani’ 3000 NaN NaN NaN 627 986 2.721861 620 1032 2.924245 637 1029 2.836656

’dixmaanj’ 300 1984 2054 0.65829 114 206 0.066094 109 211 0.065167 101 190 0.059669

’dixmaanj’ 9000 433 486 3.704114 203 342 2.639958 198 332 2.611461 198 337 2.594001

’dixmaanj’ 15000 500 553 6.82994 202 359 4.599998 183 336 4.205175 192 317 4.068118

’dixmaank’ 30000 640 703 17.35754 202 351 8.516839 189 321 7.615254 192 351 8.315288

’dixmaank’ 3000 NaN NaN NaN 135 231 0.633973 119 223 0.6079 166 275 0.731622

’dixmaank’ 30000 640 703 16.85895 202 351 8.482023 189 321 7.709402 192 351 9.147786

’dixon3dq’ 10 77 114 0.003808 60 117 0.002931 57 109 0.004382 52 106 0.003532

’dixon3dq’ 50 386 423 0.015569 458 733 0.025348 279 468 0.017681 272 445 0.018375

’eg2’ 4 32 67 0.007854 22 60 0.001858 26 67 0.002469 22 60 0.002194

’eg2’ 10 66 141 0.004638 58 118 0.003596 35 85 0.002841 58 118 0.005193

’eg2’ 50 71 206 0.007189 165 428 0.014942 120 319 0.01263 101 242 0.008975

’freuroth’ 4 218 343 0.023291 80 224 0.006497 51 159 0.005727 68 185 0.008482

’freuroth’ 10 301 395 0.019306 52 169 0.005128 71 201 0.009838 52 169 0.006593

’freuroth’ 50 290 390 0.027218 68 194 0.007977 64 188 0.010353 68 189 0.010171

’freuroth’ 100 275 351 0.026272 65 238 0.01309 63 181 0.010686 60 169 0.009422

’himmelbg’ 500 2 9 0.006417 2 9 0.000912 2 9 0.000876 2 9 0.000863

’himmelbg’ 1000 2 9 0.00218 2 9 0.001992 2 9 0.001696 2 9 0.001523

’himmelbg’ 5000 3 21 0.013903 3 21 0.012772 3 21 0.01348 3 21 0.012958

’himmelbg’ 10000 2 11 0.018961 2 11 0.017717 2 11 0.018844 2 11 0.017778



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 10 of 15

’liarwhd’ 500 88 227 0.025006 70 232 0.020955 58 226 0.021396 73 226 0.023267

’liarwhd’ 1000 164 389 0.044784 95 318 0.034766 71 254 0.027863 68 254 0.029333

’liarwhd’ 5000 144 371 0.184553 89 342 0.158965 90 334 0.15707 74 271 0.12338

’liarwhd’ 10000 145 472 0.409646 94 347 0.310964 67 257 0.224857 113 404 0.328639

’penalty1’ 500 22 119 0.178701 19 107 0.139866 20 113 0.14773 25 120 0.157519

’penalty1’ 1000 21 110 0.404508 21 111 0.400485 27 117 0.426632 22 106 0.385329

’penalty1’ 4000 23 138 6.079985 26 140 6.132693 23 137 6.028452 25 137 6.049148

’penalty1’ 10000 18 79 18.10737 18 79 18.12312 17 77 17.72842 17 79 18.25694

’tridia’ 500 1434 1497 0.116049 723 1159 0.080065 630 1018 0.07713 612 980 0.081127

’tridia’ 1000 NaN NaN NaN 968 1531 0.162034 995 1671 0.202518 923 1434 0.174033

’tridia’ 4000 NaN NaN NaN NaN NaN NaN 1549 2545 1.03679 1992 3174 1.34513

’woods’ 500 NaN NaN NaN 164 335 0.023281 165 389 0.028255 169 374 0.027434

’woods’ 1000 NaN NaN NaN 207 433 0.046658 131 285 0.031851 170 352 0.039766

’woods’ 5000 NaN NaN NaN 237 506 0.243439 124 286 0.151039 245 546 0.283006

’woods’ 10000 515 653 0.626043 176 383 0.350396 139 345 0.317029 125 287 0.276205

’bdexp’ 500 2 7 0.011732 2 7 0.002199 2 7 0.001729 2 7 0.001717

’bdexp’ 1000 2 7 0.003878 2 7 0.004103 2 7 0.004569 2 7 0.003686

’bdexp’ 5000 2 8 0.01844 3 19 0.032291 3 19 0.029448 3 19 0.028148

’bdexp’ 10000 2 9 0.042049 2 9 0.042081 2 9 0.042312 2 9 0.044224

’exdenschnf’ 500 46 103 0.019623 23 75 0.007377 22 78 0.011364 23 75 0.009551

’exdenschnf’ 1000 56 113 0.020576 25 80 0.015406 29 84 0.016264 28 85 0.020477

’exdenschnf’ 5000 43 102 0.088943 23 83 0.066448 29 90 0.078213 24 84 0.068349

’exdenschnf’ 10000 35 98 0.153769 27 90 0.141202 23 96 0.15179 27 90 0.139626

’biggsb1’ 4 17 47 0.006482 26 56 0.001628 17 49 0.001473 26 56 0.001858

’biggsb1’ 10 36 73 0.003433 35 77 0.003496 46 93 0.005471 35 77 0.003913

’biggsb1’ 100 364 396 0.017962 337 521 0.026132 231 369 0.016808 376 630 0.028446

’biggsb1’ 500 1692 1726 0.107611 1495 2408 0.148393 1128 1800 0.117736 1437 2255 0.144348

’power1’ 4 36 73 0.005881 33 80 0.002146 28 65 0.002168 33 80 0.002568

’power1’ 10 95 141 0.005946 76 128 0.003989 81 155 0.006034 74 125 0.005361

’power1’ 50 398 458 0.019117 473 778 0.028975 440 708 0.02847 460 698 0.02841

’power1’ 100 933 1001 0.040213 999 1637 0.058882 817 1326 0.057257 1030 1649 0.072432

’diagonal1’ 4 19 51 0.0055 16 46 0.001338 15 47 0.001364 16 46 0.001381

’diagonal1’ 10 31 69 0.003252 22 62 0.003238 28 65 0.002222 22 62 0.002042

’diagonal1’ 50 223 276 0.014341 50 105 0.005477 48 100 0.006636 49 104 0.004142

’diagonal1’ 100 204 258 0.017461 70 140 0.008903 73 139 0.007528 68 137 0.006523

’singx’ 100 170 366 0.029554 185 440 0.03583 150 380 0.037192 124 293 0.021211

’singx’ 500 238 502 0.798809 205 474 0.753916 152 365 0.605223 94 240 0.386519

’lin’ 10 8 40 0.150211 8 40 0.149828 8 40 0.165787 8 40 0.149329

’lin’ 100 10 51 0.362836 10 51 0.348565 10 51 0.352339 10 51 0.356541

’lin’ 500 14 55 0.604231 14 55 0.596905 14 55 0.609829 14 55 0.620677

’lin’ 1000 12 61 37.77361 12 61 37.48226 12 61 37.69565 12 61 37.33486



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’osb2’ 11 NaN NaN NaN 465 797 0.072565 362 636 0.060535 460 772 0.076487

’rosex’ 50 127 275 0.017377 64 213 0.011807 41 138 0.009282 54 179 0.011496

’rosex’ 100 114 269 0.021787 58 166 0.010547 48 165 0.010602 56 151 0.0093

’trid’ 500 154 201 0.271871 30 76 0.094568 31 76 0.097215 30 76 0.107171

’trid’ 1000 206 256 0.964584 31 77 0.285915 31 81 0.297505 31 77 0.290174

’trid’ 5000 404 457 38.05434 35 86 7.110678 34 84 6.892447 35 86 6.994909

’trid’ 10000 579 635 196.8889 76 154 47.65638 76 169 32.48472 71 149 36.09446

Table 2: Comparison of the proposed method with HS and Perry algorithms, showing
percentage improvements in NI, NF, and CPUT.

Comparison with HS

Metric HS New New (ϵm = 1)

NI 100% 39.30% 43.95%

NF 100% 51.86% 55.36%

CPUT 100% 37.72% 39.03%

Comparison with Perry

Metric Perry New New (ϵm = 1)

NI 100% 81.35% 90.98%

NF 100% 84.30% 90.00%

CPUT 100% 93.08% 96.32%

The comparison shown in Table 2 highlights the performance of the proposed method
compared to the classical HS and Perry algorithms, focusing on three key metrics (NI, NF
and CPUT):

Comparison with HS

• NI: The new method achieves an improvement of 60.70%, and with the specific case
(ϵm = 1), the improvement slightly decreases to 56.05%.

• NF: The new method shows a 48.14% improvement, which further increases to
44.64% in the specific case.

• CPUT: The time is reduced by 62.28%, and in the specific case, it is reduced by
60.97%.

Comparison with Perry

• NI: The proposed method shows a significant improvement of 18.65%, and in the
specific case, it improves further to 9.02%.

• NF: An improvement of 15.70% is observed, which reaches 10.00% in the specific
case.



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 12 of 15

• CPUT: The computational time decreases by 6.92%, and in the specific case, it
improves even further to 3.68%.

From Table 2, it is evident that the proposed method outperforms the HS and Perry
algorithms in terms of efficiency, requiring fewer iterations, fewer function evaluations,
and less computation time. The specific case (ϵm = 1) further enhances the performance,
particularly in reducing the computation time. This demonstrates the robustness and
efficiency of the new method compared to traditional approaches.

0 0.5 1 1.5 2 2.5
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ

p
s
(τ

)

 

 

HS

Perry

New

New(ε
m

=1)

(a) Performance Profile for NI.

0 0.5 1 1.5 2 2.5
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ

p
s
(τ

)

 

 

HS

Perry

New

New(ε
m

=1)

(b) Performance Profile for NF.

0 0.5 1 1.5 2 2.5
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ

p
s
(τ

)

 

 

HS

Perry

New

New(ε
m

=1)

(c) Performance Profile for CPUT.

Figure 1: Performance profiles for (a) NI, (b) NF, and (c) CPUT.



A. L. Ibrahim et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5932 13 of 15

6. Conclusion

In this paper, a new method for solving unconstrained optimization problems based on
the CG algorithm was proposed. To evaluate its efficiency, the new method was compared
with the classical HS and Perry methods using three performance metrics: the NI, NF,
and CPUT. As shown in Table 2, the new method demonstrated significant improvements
over the classical methods. Compared to the HS method, the proposed approach achieved
a 60.70% reduction in NI, 48.14% in NF, and 62.28% in CPUT. Similarly, when compared
to the Perry method, it showed improvements of 18.65% in NI, 15.7% in NF, and 6.92% in
CPUT. Furthermore, the proposed method satisfied the descent condition while ensuring
global convergence. These characteristics confirm that the method provides not only
accurate solutions but also stability and strong performance across a wide range of small-
and large-scale optimization problems. Additionally, the special case of the proposed
method (ϵm = 1) showed further performance enhancements, underscoring the flexibility
and robustness of the proposed approach in addressing diverse optimization challenges.
These results indicate that the new method is not only more efficient in terms of reducing
iterations and processing time but also serves as a competitive and reliable option for
solving unconstrained optimization problems, adding a valuable contribution to the family
of CG methods.

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