EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 3, Article Number 6145
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Modifying Spectral Conjugate Gradient Method for
Solving Unconstrained Optimization Problems

Kamal Jameel Murad1,∗, Salah Gazi Shareef1

1 Department of Mathematics, College of Science, University of Zakho, Kurdistan Region,
Iraq

Abstract. This paper proposes a new spectral conjugate gradient method for large-scale uncon-
strained optimization, designed to improve convergence efficiency by reducing both iteration counts
and function evaluations. The method introduces a modified spectral coefficient and a new search
direction formula that guarantees descent and sufficient descent conditions at every iteration, with-
out increasing the per-iteration computational burden. Unlike existing methods such as the clas-
sical conjugate gradient algorithm, the proposed scheme integrates spectral scaling in a way that
enhances direction quality and step stability. Theoretical analysis establishes global convergence
under standard assumptions. Extensive numerical experiments on a diverse set of test problems
demonstrate the superior performance of the proposed method over the classical conjugate gradi-
ent and spectral conjugate gradient methods, particularly in scenarios where fast convergence and
low evaluation cost are critical. These results suggest that the proposed method offers a robust
and computationally efficient alternative for solving unconstrained optimization problems. Future
work will explore its application to structured and real-world large-scale problems.

2020 Mathematics Subject Classifications: 65K05, 46N10, 90C26, 90C30
Key Words and Phrases: Unconstrained Optimization, Spectral Conjugate Gradient, Global
Convergence, Descent Conditions

1. Introduction

This paper addresses the unconstrained optimization problem:

min{f(x) : x ∈ Rn}, (1)

where the objective function f : Rn → R has continuous partial derivatives, and its gradi-
ent ∇f(x) = g(x) is available for evaluation. Unconstrained optimization plays a crucial
role in a wide array of applications, including machine learning [1–3], image processing [4–
6], and portfolio optimization, as well as in fields such as economics, engineering, manage-
ment science, and industrial applications [7–11]. Various methods have been proposed for

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6145

Email addresses: kamal.murad@uoz.edu.krd (K. J. Murad),
salah.shareef@uoz.edu.krd (S. G. Shareef)

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



K. J. Murad, S. G. Shareef / Eur. J. Pure Appl. Math, 18 (3) (2025), 6145 2 of 17

solving problem (1), including Newton-type methods, quasi-Newton methods, spectral gra-
dient methods, and conjugate gradient (CG) methods [12–14]. In addition, derivative-free
optimization techniques such as the Nelder–Mead simplex method, generalized simulated
annealing, and genetic algorithms are also used [15–17]. This paper specifically focuses
on CG methods and introduces a new spectral conjugate gradient method designed to
efficiently tackle large-scale instances of problem (1).

Recently, geometric optimization approaches have garnered increasing interest. The
application of Riemannian geometry to optimization problems, particularly on matrix
manifolds, has led to the development of structure-preserving methods. For example, Arif
et al. [18] proposed an Extended Hamiltonian Algorithm (EHA) for solving the Algebraic
Lyapunov Equation on the manifold of positive-definite Hermitian matrices. This ap-
proach utilizes a geodesic-based cost function and exhibits superior convergence compared
to traditional gradient-based methods. In addition, studies on the geometry of statistical
manifolds, such as the Freund manifold [19] and Gamma exponential manifold [20], have
provided insights into the role of geodesic instability and Jacobi fields in understanding
optimization paths in curved spaces. These geometric insights inspire us to enhance clas-
sical CG methods by incorporating curvature-aware spectral adjustments to improve both
convergence and robustness, particularly in high-dimensional and ill-conditioned settings.
Motivated by such developments, we propose a new spectral conjugate gradient method
tailored to large-scale optimization problems. Our method incorporates adaptive spec-
tral parameters, while maintaining the simplicity and memory efficiency characteristic of
traditional CG schemes.

CG methods are widely recognized for their simplicity, efficiency, and low memory
requirements, making them highly suitable for large-scale optimization problems. The
standard iteration scheme of classical CG methods is given by:

xk+1 = xk + αkdk, (2)

where dk denotes the search direction, which is defined as:

dk+1 =

{
−g0, k = 0,

−gk+1 + βkdk, k ≥ 1.
(3)

Here, gk = g(xk), βk is the conjugate parameter, and αk is the step size, which can be
computed using exact or inexact line search techniques. Due to the computational cost
associated with exact line search, inexact line searches such as the Wolfe conditions are
often employed: {

f(xk + αkdk) ≤ f(xk) + δαkg
T
k dk,

g(xk + αkdk)
Tdk ≥ σgTk dk,

(4)

or the strong Wolfe conditions:{
f(xk + αkdk) ≤ f(xk) + δαkg

T
k dk,

|g(xk + αkdk)
Tdk| ≤ σ|gTk dk|.

(5)



K. J. Murad, S. G. Shareef / Eur. J. Pure Appl. Math, 18 (3) (2025), 6145 3 of 17

In these conditions, the parameters δ and σ are chosen such that 0 < δ < σ < 1.
Different choices of the parameter βk lead to various CG methods, the most well-known

of which include Hestenes-Stiefel (HS) [21], Fletcher-Reeves (FR) [22], Polak-Ribière-
Polyak (PRP) [23, 24], Liu-Storey (LS) [25], Dai-Yuan (DY) [26], and Conjugate-Descent
(CD) [27]. For example:

βHS
k =

gTk+1yk

yTk dk
, (6)

βFR
k =

∥gk+1∥2

∥gk∥2
, (7)

βPRP
k =

gTk+1yk

∥gk∥2
, (8)

βLS
k =

gTk+1yk

−gTk dk
, (9)

βDY
k =

∥gk+1∥2

yTk dk
, (10)

βCD
k =

∥gk+1∥2

−gTk dk
, (11)

where yk = gk+1 − gk and ∥ · ∥ denotes the Euclidean norm.
Theoretically, for strongly convex quadratic functions, all choices of βk are equivalent

when exact minimization is applied. However, for non-quadratic objective functions, the
performance varies significantly depending on the choice of βk [28]. Among these methods,
the PRP method has been shown to be more computationally efficient than the FR method
and has global convergence properties under exact line search when the objective function
is convex [24]. Several variations of CG methods have been proposed to improve their
performance further [29–34].

Building on the spectral gradient approach [35], Bergin et al. [36] introduced the
spectral conjugate gradient (B-SCG) method, where the search direction is defined by:

dk+1 =

{
−g0, k = 0,

−θkgk+1 + βkdk, k ≥ 1.
(12)

Here, θk is the spectral parameter, and βk is the conjugate parameter defined as:

θk =
vTk vk

vTk yk
, βk =

(θkyk − vk)
T gk+1

dTk yk
. (13)

The SCG method reduces to the classical CG method when θk = 1 and to the spectral
gradient method when βk = 0. The SCG [37] was modified by Yu et al. [38] in order to
achieve the descent directions. Other modifications based on different descent conditions
have been suggested by Wan et al. [39] and Zhang et al. [40], focusing on PRP and FR



K. J. Murad, S. G. Shareef / Eur. J. Pure Appl. Math, 18 (3) (2025), 6145 4 of 17

variants. Moreover, leveraging the strong convergence of the Newton method, Andrei [41]
introduced an accelerated CG approach that integrates Newton’s method to improve CG
performance.

Recently, modern CG variants and alternative optimization methods have been pro-
posed to enhance robustness and efficiency, particularly for large-scale or noisy problems.
Hybrid CG methods combine conjugate directions with quasi-Newton updates or restart
strategies [28, 42], while stochastic CG techniques address gradient uncertainty in data-
driven applications [43]. Moreover, quasi-Newton approaches such as the limited-memory
BFGS (L-BFGS) method [44] are widely used due to their fast convergence and low mem-
ory requirements. Compared to these methods, our approach preserves the simplicity
and memory efficiency of classical CG methods while incorporating spectral features that
improve convergence behavior without relying on Hessian approximations or stochastic
estimates.

In this paper, we propose a novel spectral conjugate gradient method that incorporates
a modified gradient and an alternative search direction, as given by Eq. (12). Our method
ensures that the descent and sufficient descent properties are satisfied per iteration, with-
out adding computational overhead. Under appropriate assumptions, we establish the
global convergence of the proposed method.

The remainder of this paper is organized as follows. In Section 2, we introduce the
proposed spectral conjugate gradient algorithm. Section 3 presents the analysis of the suf-
ficient descent property and the global convergence of the method. In Section 4, we provide
numerical experiments that compare the proposed method with the HS-CG and B-SCG
methods. Finally, Section 5 summarizes the main contributions and outlines directions for
future research.

2. The New Spectral Conjugate Gradient Algorithm

In this section, we introduce a new spectral conjugate gradient (New-SCG) method to
solve unconstrained optimization problems of the form (1), utilizing a modified gradient-
difference vector inspired by [45].

We define the adjusted gradient-difference vector as follows:

y∗k = yk +
(0.2− ρk)

(1− ρk)
·
∥vk∥ − 2

√
ϵm(1 + ∥xk+1∥)

2
√
ϵm(1 + ∥xk+1∥)

· yk, (14)

where vk = xk+1 − xk, and yk = gk+1 − gk. Here, 0.2 < ρk < 1 and ϵm represents machine
precision, typically around 10−16.

The new search direction is given by:

dNew-SCG
k+1 =

{
−g1, k = 0,

−θkgk+1 + βkdk, k ≥ 1.
(15)

where θk is a spectral scaling parameter, and βk is a conjugate gradient parameter.



K. J. Murad, S. G. Shareef / Eur. J. Pure Appl. Math, 18 (3) (2025), 6145 5 of 17

Dai and Liao [46] proposed the classical conjugacy condition:

dTk+1yk = −tgTk+1vk, where t ≥ 0, (16)

which we modify by substituting yk with y∗k, yielding:

dTk+1yk =
−tgTk+1vk

1 +M1M2
, (17)

with:

M1 =
0.2− ρk
1− ρk

, M2 =
∥vk∥ − 2

√
ϵm(1 + ∥xk+1∥)

2
√
ϵm(1 + ∥xk+1∥)

. (18)

Multiplying both sides of Eq. (15) by yk and using Eq. (17), we define the spectral
parameter as:

θk =
βkd

T
k yk +

tgTk+1vk
1+M1M2

gTk+1yk
. (19)

In particular, if we take βk = βHS
k (Hestenes-Stiefel formula), then:

θk = 1 +
tgTk+1vk

(1 +M1M2)gTk+1yk
. (20)

Remark 1. When t = 0, the proposed spectral parameter in Eq. (20) reduces to θk = 1,
which implies that the search direction becomes:

dNew-SCG
k+1 = −gk+1 + βHS

k dk, (21)

i.e., the method reduces to the classical HS conjugate gradient method. Therefore, the
proposed algorithm can be seen as a spectral extension of the HS method.

Based on the above analysis, we propose the following algorithm:



K. J. Murad, S. G. Shareef / Eur. J. Pure Appl. Math, 18 (3) (2025), 6145 6 of 17

Algorithm 1 New Spectral Conjugate Gradient Algorithm

1: Input: Initial guess x0, tolerance ϵ, parameters t ≥ 0, 0.2 < ρk < 1
2: Compute g0 = ∇f(x0), set d0 = −g0, k = 0
3: while ∥gk∥ > ϵ do
4: If ∥gk∥ = 0, then stop; otherwise, proceed to Step 5.
5: Compute the step size αk by minimizing f(xk + αkdk).
6: xk+1 = xk + αkdk
7: gk+1 = ∇f(xk+1)
8: yk = gk+1 − gk, vk = xk+1 − xk
9: M1 =

(0.2−ρk)
(1−ρk)

10: M2 =
∥vk∥−2

√
ϵm(1+∥xk+1∥)

2
√
ϵm(1+∥xk+1∥)

11: βk =
gTk+1yk

yTk dk

12: θk = 1 +
tgTk+1vk

(1+M1M2)gTk+1yk

13: dk+1 = −θkgk+1 + βkdk
14: If ∥gk+1∥2 ≤ |gTk gk+1|/0.2, go to Step 4; otherwise, set k = k + 1 and repeat Step

5.
15: end while
16: Output: xk

3. Algorithm Characteristics

This section discusses the descent properties, sufficient descent conditions, and global
convergence of the proposed algorithm.

Theorem 1: If the search direction dk+1 is generated by equation (15), where βk is
defined by equation (6) and the spectral parameter θk is given by equation (20), then:

gTk+1dk+1 ≤ 0. (22)

Proof: By multiplying both sides of equation (15) by gk+1, we obtain:

gTk+1dk+1 = −

(
1 +

tgTk+1vk

(1 +M1M2)gTk+1yk

)
gTk+1gk+1 +

gTk+1yk

dTk yk
gTk+1dk. (23)

Simplifying this expression, we get:

gTk+1dk+1 = −gTk+1gk+1 +
gTk+1yk

dTk yk
gTk+1dk −

tgTk+1vkg
T
k+1gk+1

(1 +M1M2)gTk+1yk
. (24)

In the case of an exact search direction, the descent condition is satisfied:

gTk+1dk+1 = −∥gk+1∥2 ≤ 0. (25)



K. J. Murad, S. G. Shareef / Eur. J. Pure Appl. Math, 18 (3) (2025), 6145 7 of 17

For an inexact search direction (gTk+1dk ̸= 0), we use the following inequalities:

gTk+1vk ≤ vTk yk, gTk+1yk ≤ ∥gk+1∥∥yk∥. (26)

From these, we derive:

gTk+1dk+1 ≤ −
(
1− ∥yk∥

∥gk+1∥

)
∥gk+1∥2 −

(
tgTk+1vk

(1 +M1M2)gTk+1yk

)
gTk+1gk+1. (27)

Since 0 ≤ ∥yk∥
∥gk+1∥ ≤ 1 and using the inequalities in (26), we can rewrite equation (27)

as:

gTk+1dk+1 ≤ −
tvTk yk∥gk+1∥

(1 +M1M2)∥yk∥
. (28)

Since t, vTk yk, ∥gk+1∥, and ∥yk∥ are non-negative, and M1 < 0, M2 < 0 imply that
M1M2 > 0, it follows that:

gTk+1dk+1 ≤ 0. (29)

This completes the proof.

Theorem 2: The search direction dk+1, defined by equation (15), where βk is from
equation (6) and the spectral parameter θk is given by equation (20), satisfies:

gTk+1dk+1 ≤ −µ∥gk+1∥2, (30)

where µ > 0 is a positive constant.

Proof: From Theorem 1 and equation (28), we have:

gTk+1dk+1 ≤ −
[

tvTk yk
(1 +M1M2)∥yk∥∥gk+1∥

]
∥gk+1∥2. (31)

Let:

µ =
tvTk yk

(1 +M1M2)∥yk∥∥gk+1∥
> 0. (32)

This completes the proof.
The global convergence of the algorithm is guaranteed under the following assumptions:

Assumptions I

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

• The function f is continuously differentiable in some neighborhood N and its gra-
dient satisfies the Lipschitz condition:

∥g(x)− g(y)∥ ≤ L∥x− y∥, ∀x, y ∈ S. (33)



K. J. Murad, S. G. Shareef / Eur. J. Pure Appl. Math, 18 (3) (2025), 6145 8 of 17

• There exists a constant b > 0 such that:

∥g(x)∥ ≤ b, ∀x ∈ S. (34)

Using these assumptions, we derive the following result:

Lemma 1: [46] Let Assumptions (I) hold. Consider the 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
= ∞, (35)

then it follows that
lim
k→∞

inf ∥gk+1∥ = 0. (36)

Theorem 3: Assuming that Assumptions (I) hold, and the sequences {xk}, {dk},
{gk}, and {αk} are generated by Algorithm 1, we conclude that

lim
k→∞

inf ∥gk+1∥ = 0. (37)

Proof: Starting from equation (15) and using the parameters βk from equation (6)
and the spectral parameter θk from equation (20), we obtain the following expression:

∥dk+1∥ = ∥ −

(
1 +

tgTk+1vk

(1 +M1M2)gTk+1yk

)
gk+1 +

gTk+1yk

dTk yk
dk∥. (38)

This leads to the following bound on ∥dk+1∥:

∥dk+1∥ ≤

∣∣∣∣∣1 + tgTk+1vk

(1 +M1M2)gTk+1yk

∣∣∣∣∣ ∥gk+1∥+

∣∣∣∣∣gTk+1yk

dTk yk

∣∣∣∣∣ ∥dk∥. (39)

From equation (26), we have:

∥dk+1∥ ≤
∣∣∣∣1 + t

(1 +M1M2)

∣∣∣∣ ∥vk∥+ ∣∣∣∣∥gk+1∥∥yk∥
dTk yk

∣∣∣∣ ∥dk∥. (40)

Using equation (34) and the Lipschitz condition ∥yk∥ ≤ L∥vk∥, along with the fact
that yTk vk ≥ ϑ∥vk∥2, we obtain:

∥dk+1∥ ≤
∣∣∣∣1 + t

(1 +M1M2)

∣∣∣∣ ∥vk∥+ αkbL∥vk∥
ϑ∥vk∥2

∥dk∥, (41)

which simplifies to:

∥dk+1∥ ≤
∣∣∣∣1 + t

(1 +M1M2)

∣∣∣∣ ∥vk∥+ bL

ϑ
. (42)



K. J. Murad, S. G. Shareef / Eur. J. Pure Appl. Math, 18 (3) (2025), 6145 9 of 17

Since ∥vk∥ = ∥x−xk∥, we define D = max{∥x−xk∥} for all x, xk ∈ R. Thus, equation
(35) becomes:

∥dk+1∥ ≤ b+
tD

(1 +M1M2)
+

bL

ϑ
= ϕ. (43)

It follows that: ∑
k≥1

1

∥dk+1∥2
≥
∑
k≥1

1

ϕ2
= ∞. (44)

By applying Lemma 1, we conclude that:

lim
k→∞

inf ∥gk+1∥ = 0, (45)

which completes the proof.

4. Numerical Results for Unconstrained Optimization

This section presents the results of implementing the New-SCG algorithm for solving
unconstrained optimization problems. A comparative analysis is conducted between the
proposed algorithm and the classical HS and B-SCG methods. The tests were carried
out using well-established benchmark functions [47] with varying problem dimensions.
All codes were developed in FORTRAN 95, and the line search routine employed cubic
interpolation based on both function and gradient evaluations. The letter ’F’ in the tables
indicates that the method failed to locate the minimum.

Table 1 reports the performance of the algorithms in terms of the number of iterations
(NOI) and the number of function evaluations (NOF). The summary in Table 2 further
confirms that the New-SCG algorithm outperforms both HS and B-SCG methods with
respect to NOI and NOF metrics.

All tests were initialized from standard starting points, and the comparative numerical
results are visualized in Figures 1a and 1b. The relative performance of the New-SCG,
HS, and B-SCG methods is assessed using performance profiles in accordance with the
approach described in [48].

Let S = 2 represent the set of solvers being compared, and p = 72 denote the total
number of test problems. Let lp,s represent the number of objective function evaluations
required by solver s to solve problem p. The performance ratio is defined as

rp,s =
lp,s
l∗p

,

where l∗p is the minimum number of evaluations among all solvers:

l∗p = min{lp,s : s ∈ S}.

It is evident that rp,s ≥ 1 for all p and s. If a solver fails to solve a problem, the ratio rp,s
is assigned a large number M .



K. J. Murad, S. G. Shareef / Eur. J. Pure Appl. Math, 18 (3) (2025), 6145 10 of 17

The performance profile for each solver s is given by the cumulative distribution function
of the performance ratio rp,s:

pτ (τ) =
|{p ∈ P : rp,s ≤ τ}|

np
.

Here, ps(1) represents the percentage of problems for which solver s is the best.

Table 1: Comparison of Performance Metrics (NOI and NOF) for Different Methods Across
Various Test Functions and Dimensions

Method Classical HS B-SCG New-SCG

Test Function Dimensions NOI NOF NOI NOF NOI NOF

Wolfe

4 11 24 11 29 11 25
10 32 65 32 71 32 65
100 49 99 49 100 44 89
500 52 105 52 107 46 93
1000 70 141 60 122 50 101
5000 165 348 168 338 129 270

G-Central

4 22 159 16 77 21 147
10 22 159 16 103 22 160
100 22 159 19 103 22 160
500 23 171 28 153 22 160
1000 23 171 34 186 22 160
5000 28 248 52 287 26 214

Non-diagonal

4 24 64 30 223 24 64
10 26 72 29 238 26 72
100 29 79 27 285 29 79
500 F F 27 335 29 79
1000 29 79 30 385 29 79
5000 30 81 30 420 30 81

Powell

4 37 102 47 231 34 93
10 37 102 47 231 34 93
100 40 117 47 231 34 93
500 44 136 47 231 34 93
1000 44 136 51 254 34 93
5000 44 136 55 280 34 93

Rosen

4 30 83 30 83 16 44
10 30 83 30 83 16 44
100 30 83 30 83 17 47
500 30 83 30 83 17 47
1000 30 83 30 83 17 47
5000 30 83 30 83 17 47

Miele

4 28 85 40 138 28 89



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10 31 102 43 150 28 89
100 33 114 49 186 28 89
500 40 146 55 218 29 102
1000 46 176 55 218 29 102
5000 54 211 61 247 30 105

Wood

4 30 68 30 68 26 60
10 30 68 30 68 26 60
100 30 68 30 68 26 60
500 30 68 30 68 26 60
1000 30 68 30 68 26 60
5000 30 68 30 68 27 62

Sum

4 3 11 3 11 3 11
10 6 34 6 26 6 28
100 14 81 17 88 15 89
500 21 124 28 162 20 104
1000 23 128 42 199 25 139
5000 31 159 43 227 28 117

Edger

4 5 14 5 14 5 14
10 5 14 5 14 5 14
100 5 14 5 14 5 14
500 6 16 6 16 5 14
1000 6 16 6 16 5 14
5000 6 16 6 16 6 17

Shallow

4 8 21 8 21 8 21
10 8 21 8 21 8 21
100 8 21 8 21 8 21
500 8 21 8 21 8 21
1000 9 24 9 24 8 21
5000 9 24 9 24 8 21

Cubic

4 12 35 22 141 12 35
10 13 37 22 141 13 37
100 13 37 22 141 13 37
500 13 37 22 141 13 37
1000 13 37 22 141 13 37
5000 13 37 22 141 13 37

Beale

4 11 28 11 28 11 28
10 11 28 11 28 11 28
100 12 30 12 30 12 30
500 12 30 12 30 12 30
1000 12 30 12 30 12 30
5000 12 30 12 30 12 30

Total 1852 5927 2091 9040 1570 4967



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Table 2: Comparing the Rate of Improvement between the New-SCG Method and the
Classical HS and B-SCG Methods.

Methods NOI (%) NOF (%)

Classical HS 100.00 100.00

New-SCG 84.77 83.80

B-SCG 100.00 100.00

New-SCG 75.08 54.94

Rate of Improvement (%)

Compared with NOI NOF

Classical HS 15.23 16.20

B-SCG 24.92 45.06

The table 2 compares the performance of the proposed New-SCG algorithm with both
the classical HS and the B-SCG methods in terms of the NOI and the NOF. The analysis
of the data reveals the following improvements:

• Compared with Classical HS:

– Improvement in NOI: The New-SCG algorithm achieves a 15.23% reduction
in the NOI, indicating faster convergence.

– Improvement in NOF: There is a 16.20% reduction in the NOF, suggesting
higher computational efficiency.

– Overall Average Improvement: The mean of both improvements (NOI
and NOF) is approximately 15.72%, demonstrating the New-SCG method’s
significant advantage over the classical HS method.

• Compared with B-SCG:

– Improvement in NOI: The New-SCG method reduces the NOI by 24.92%,
showing a more pronounced convergence speedup.

– Improvement in NOF: The reduction in function evaluations reaches 45.06%,
highlighting a substantial gain in computational cost.

– Overall Average Improvement: The average of both metrics yields an over-
all improvement of approximately 34.99%, clearly indicating that the New-SCG
method outperforms the B-SCG method in both convergence rate and compu-
tational efficiency.

These results confirm that the proposed New-SCG algorithm offers consistent and
notable improvements over both classical HS and B-SCG methods. Its reduced number of
iterations and function evaluations collectively contribute to faster and more cost-effective
optimization performance.



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0 0.2 0.4 0.6 0.8 1 1.2
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

τ

p
s
(τ

)

 

 

Classical HS

B−SCG

New−SCG

(a) Performance profiles based on the NOI.

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

)

 

 

Classical HS

B−SCG

New−SCG

(b) Performance profiles based on the
NOF.

Figure 1: Performance profiles of the New method.

5. Conclusion

This paper proposed a novel search method, the New-SCG algorithm, designed to en-
hance the efficiency of large-scale unconstrained optimization problems. The theoretical
properties of the method were thoroughly analyzed, including proofs for both descent and
sufficient descent conditions and global convergent, establishing its robustness. Numerical
evaluations demonstrated that the New-SCG algorithm significantly outperforms classical
optimization methods such as HS and B-SCG methods. Specifically, the New-SCG algo-
rithm achieved a 15.23% reduction in the NOI and a 16.20% reduction in the NOF when
compared to the HS method. In comparison to B-SCG, the New-SCG method showed even
more substantial improvements, with reductions of 24.92% in NOI and 45.06% in NOF.
These findings underscore the superior efficiency of the New-SCG method in reducing
computational costs while enhancing convergence speed. To provide a more comprehen-
sive assessment, future studies will expand the evaluation of the New-SCG algorithm by
applying it to non-convex and non-smooth optimization problems, which are typically
more complex both theoretically and computationally. Furthermore, comparisons will be
made with other optimization techniques widely used in real-world applications, such as
in engineering design, control systems, and machine learning. This broader scope will help
evaluate the algorithm’s general applicability in solving diverse, real-world optimization
problems. While this study focuses primarily on the theoretical and numerical analysis
of the New-SCG algorithm, future research will explore its practical implications, espe-
cially in engineering fields. For example, applications in structural optimization, process



K. J. Murad, S. G. Shareef / Eur. J. Pure Appl. Math, 18 (3) (2025), 6145 14 of 17

control, and large-scale machine learning models could benefit from the improved com-
putational efficiency and reduced cost associated with this method. In addition to these
practical considerations, future work will address the limitations of the proposed algo-
rithm. In particular, further analysis is needed on the algorithm’s performance in the
presence of ill-conditioned optimization problems, where traditional methods may strug-
gle. Future versions of the New-SCG algorithm will also seek to extend its applicability
to constrained optimization problems and non-smooth objective functions, ensuring that
it can handle a broader class of problems encountered in engineering and industrial ap-
plications. In summary, the New-SCG algorithm presents a promising advancement in
large-scale unconstrained optimization. With its superior computational efficiency and
theoretical foundation, it holds significant potential for a wide range of practical applica-
tions, while offering several avenues for future enhancement and refinement.

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