A Quadratic Model based Conjugate Gradient Optimization Method

ISAM H. HALIL1, ISSAM A.R. MOGHRABI2, AHMED A. FAWZE3, BASIM A. HASSAN3,

HISHAM M. KHUDHUR3

1Department of Mathematics, College of Science,

Kirkuk University, Kirkuk,

IRAQ

2Department of Computer Science, College of Arts and Sciences,

University Central Asia, Naryn,

KYRGYZ REPUBLIC

3Department of Mathematics, College of Computer Science and Mathematics,

University of Mosul, Mosul,

IRAQ

Abstract: - In this paper, we introduce a nonlinear scaled conjugate gradient method, operating on the premise

of a descent and conjugacy relationship. The proposed algorithm employs a conjugacy parameter that is

determined to ensure that the method generates conjugate directions. It also utilizes a parameter that scales the

gradient to enhance the convergence behavior of the method. The derived method not only exhibits the crucial

feature of global convergence but also maintains the generation of descent directions. The efficiency of the

method is established through numerical tests conducted on a variety of high-dimensional nonlinear test

functions. The obtained results attest to the improved behavior of the derived algorithm and support the

presented theory.

Key-Words: - unconstrained optimization, conjugate gradient methods, line search methods, global

convergence, quadratic modelling, non-linear programming.

Received: March 13, 2023. Revised: October 29, 2023. Accepted: November 13, 2023. Published: December 6, 2023.

1 Introduction

Our concern in this paper is problems of the form:

𝑚𝑖𝑛 (𝑥), where 𝑥∈𝑅 , (1)

for which 𝑓 is a differentiable convex function. For

large dimension n, conjugate gradient (CG) methods

are among the most sophisticated and

straightforward approaches proposed to solve (1),

due to storage considerations. Starting with a guess

𝑥∈𝑅, the CG method produces the sequence

󰇝𝑥󰇞 using the recurrence:

𝑥 𝑥𝛼𝑑, for 𝑘0,1,… (2)

where 𝛼 is a positive scalar representing the step

length along the search direction 𝑑, which is

calculated using some line search method. The

search vectors are derived using:

𝑑 𝑔 if k0

𝑔𝛽𝑑 if k0.

󰇛3󰇜

The parameter 𝛽 in (3) is referred to as the

conjugacy parameter and 𝑔 is the gradient vector

evaluated at 𝑥. The primary factor in CG

strategies is that they generate search directions 𝑑,

which are downhill. The particular choice of the

scalar parameter 𝛽 defines different and new

conjugate gradient algorithms. For convergence

purposes and for aiding in ensuring that the

generated search directions are downhill, the step

length 𝛼 in (2) is computed subject to satisfying the

strict conditions, [1,2], given by:

𝑓󰇛𝑥𝛼𝑑󰇜𝑓󰇛𝑥󰇜𝛿𝛼𝑔

𝑑 󰇛4󰇜

𝑑

𝑔󰇛𝑥𝛼𝑑󰇜𝜎𝑑

𝑔 , 󰇛5󰇜

where 0𝛿𝜎 to guarantee that 𝑑 is a

downward direction.

For a quadratic convex problem with positive

definite A of the form:

𝑓󰇛𝑥󰇜

𝑥𝐴𝑥𝑏𝑥𝑐 (6)

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.101

Isam H. Halil, Issam A.R. Moghrabi,

Ahmed A. Fawze, Basim A. Hassan, Hisham M. Khudhur

E-ISSN: 2224-2880

925

Volume 22, 2023

and for accurate line searches used to solve:

𝛼𝑎𝑟𝑔𝑚𝑖𝑛



𝑓

󰇛𝑥𝛼𝑑󰇜 , 󰇛7󰇜

the following conjugacy condition holds:

𝑑

𝐴

𝑑0, 󰇛8󰇜

for any 𝑖𝑗. The CG techniques, typically used

to solve systems of linear equations, are derived

with satisfying conditions (8), as is the case with the

original algorithm in, [6]. Almost all CG algorithms,

such as those in, [3], [4], [5], [6], [7], [8], [9], satisfy

(8) for quadratic objective functions and positive

definite Hessian matrix.

In this paper, we provide new variants of the

CG relations in (2) and (3) that are similar

conceptually to those in, [10], [11], [12], that

always satisfy: 𝑔

𝑑 0. 󰇛9󰇜

By the mean value theorem, for a nonlinear function

𝑓 there is a value 𝜉 such that:

𝑑

𝑦𝛼𝑑

∇𝑓󰇛𝑥𝜉𝛼𝑑󰇜𝑑. (10)

As a result of this, the following conjugacy criterion

appears to be an appropriate substitute for (8):

𝑑

𝑦0. 󰇛11󰇜

Using 󰇛3󰇜 and (11) leads to the relation (upon pre-

multiplying both sides by 𝛼):

𝛽𝑣

𝑦𝑔

𝑦0, 󰇛12󰇜

for 𝑣𝑥𝑥 or, equivalently, the conjugacy

parameter, [13]:

𝛽

 





. (13)

This paper has the following outline. We

provide a novel scaled conjugate gradient approach

in subsection 2. In Section 3, the proof that the

suggested algorithm generates a search direction

that complies with the descent requirements at each

iteration is provided. The new CG-methods' global

convergence property is proven in Section 4. The

effectiveness of the suggested CG method is

established with some numerical findings in Section

5. The final Section offers some conclusions and

future research suggestions.

2 New Scaled CG Method

We focus attention in this paper on solving

unconstrained minimization problems using the

recurrence: 𝑥 𝑥𝛼𝑑 , 󰇛14󰇜

where 𝛼0 is acquired to satisfy the conditions

for line search in (4) and (5), and 𝑑 is some

computed search direction using:

𝑑  𝜙𝑔𝛽𝑣, 󰇛15󰇜

for some scalars 𝜙 and 𝛽, whose derivation

specifies the fingerprints of the CG method.

The scalar parameters 𝜙 and 𝛽 in Equation

(15) are obtained in our approach for all 0k

based on the descent condition:

𝑔

𝑑 𝜙𝑔

𝑔

𝛽𝑔

𝑣0. 󰇛16󰇜

From 󰇛15󰇜 and (16), we get:

𝑔

𝑑 𝜙𝑔

𝑔𝛽𝑔

𝑣𝛿0,

which gives:

𝑔

𝑑 𝜙𝑔

𝑔

𝛽𝑔

𝑣𝛿. 󰇛17󰇜

Now, using the conjugacy condition 󰇛11󰇜 also

gives:

𝑦

𝑑 𝜙𝑦

𝑔𝛽𝑦

𝑣0. (18)

Let us define:

𝜔‖𝑔‖𝑦

𝑣. (19)

Assuming that 𝜔0, if using (17) and (18)

leads to:

𝜙𝛿𝑦

𝑣

𝜔

and 𝛽



. (20)

Therefore, from 󰇛15󰇜 and 󰇛20󰇜 we obtain:

𝑑

  

‖‖𝑔󰇛

󰇜

‖‖󰇛

󰇜𝑣,

or, equivalently,

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.101

Isam H. Halil, Issam A.R. Moghrabi,

Ahmed A. Fawze, Basim A. Hassan, Hisham M. Khudhur

E-ISSN: 2224-2880

926

Volume 22, 2023

𝑑

 𝛿

‖𝑔‖󰇩𝑔𝑦

𝑔

󰇛𝑦

𝑣󰇜𝑣󰇪

𝛾𝑔



󰇛

󰇜𝑣. (21)

3 The Descent Feature of the

Algorithm

The following is a presentation of the scaled

conjugate gradient (CG) approach.

Step one. Choose 𝑥∈𝑅 and the line search

parameters 𝛿 and 𝛿 such that 0𝛿𝛿1.

Compute 𝑓󰇛𝑥󰇜 and 𝑔. Take 𝑑𝑔 and 𝛼

1/‖𝑔‖.

Step two. Check if the looping can be continued. If

‖𝑔‖10, we will cease.

Step three. Update the variables 𝑥 𝑥𝛼𝑑

and compute 𝛼 0 that satisfies the

line search criteria (4) and (5).

Step four. Compute the parameter 𝛽 using

equation (13).

Step Five. Compute 𝑑 𝛾󰇛𝑔𝛽𝑑󰇜. If

the restart strategy in, [5], namely 𝑔

𝑔

0.2‖𝑔‖, is met, then use 𝑑 𝛾𝑔.

Else, define 𝑑 using (21). Compute 𝛼

‖‖

‖‖. Set 𝑘𝑘1, and continue to step 2.

The following next theorem establishes that the

search vectors generated by the derived scaled CG

technique are guaranteed to meet the descent

property, for Wolfe conditions-satisfied line

searches.

Theorem 1

Assuming 𝛼 meets conditions (3) and (4), hence,

𝑑 given by󰇛21󰇜 is a downhill direction.

Proof: Since 𝑑𝑔, we have 𝑔

𝑑

‖𝑔‖0. Multiplying 󰇛21󰇜 by 𝑔

, we have

𝑔

𝑑 

∥



∥

𝑔

𝑔





𝑔

𝑣



∥



∥



𝑔

𝑔𝑦

𝑣

𝑦

𝑔𝑦

𝑣𝑔

𝑣.

󰇛22󰇜

Applying ( 𝑢𝑣

󰇛‖𝑢‖‖𝑣‖󰇜 to (22) with

𝑢󰇛𝑦

𝑣󰇜𝑔 and 𝑣󰇛𝑔

𝑣󰇜𝑦, we get:

𝑔

𝑑

𝛿

|𝑔|󰇛𝑦

𝑣󰇜󰇩|𝑔|𝑦

𝑣

1

2󰇣|𝑔|𝑦

𝑣𝑔

𝑣󰇛|𝑦|󰇜󰇤󰇪

󰇛23󰇜

𝑔

𝑑 

∥



∥



󰇣

∥

𝑔|𝑦

𝑣

𝑔

𝑣󰇛∣𝑦∥󰇜󰇤.

󰇛24󰇜

The last term in equation (24) tends to zero

closer to the minimum, so 󰇛24󰇜 can be expressed as:

𝑔

𝑑 

∥|

󰇣

|𝑔|𝑦

𝑣󰇤



.

As a result, the direction 𝑑 meets the criteria for

a descent: 𝑔







𝑑  0.

4 Convergence Analysis

Using similar approaches to those in, [5], [14], [15],

we assume that the objective function 𝑓 satisfies the

Lipschitz continuity criterion on the level set L and

is also substantially convex. So,

𝐿󰇝𝑥∈R:𝑓󰇛𝑥󰇜𝑓󰇛𝑥󰇜󰇞.

Assumption A: There are values of 𝜇0 and 𝐿

such that, [11]:

󰇛∇𝑓󰇛𝑥󰇜∇𝑓󰇛𝑦󰇜󰇜󰇛𝑥𝑦󰇜𝜇∥𝑥𝑦|

and 󰇛∇𝑓󰇛𝑥󰇜∇𝑓󰇛𝑦󰇜󰇜𝐿∥𝑥𝑦∥

from 𝐿, for all 𝑥 and 𝑦.

Lemma 1:

Let us assume that Assumption A is true. Then

consider the general conjugate gradient approach in

(2) and (3), where 𝑑 is a downhill direction and 𝛼

is computed to satisfy the line search conditions (4)

and (5). If ∑

 

∥



∥

∞.

Then it follows that

𝑙𝑖𝑚

→ 𝑖𝑛𝑓∥𝑔∣0. (31)

Similar details can be found in [16], [17].

Theorem 2:

Assume that Assumption A holds, then 󰇝𝑥󰇞

is computed by the Algorithm (3.1) with 𝛾0.

Then 𝑙𝑖𝑚

→𝑖𝑛𝑓‖𝑔‖0. (32)

Proof:

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.101

Isam H. Halil, Issam A.R. Moghrabi,

Ahmed A. Fawze, Basim A. Hassan, Hisham M. Khudhur

E-ISSN: 2224-2880

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Volume 22, 2023

We assume the conclusion is false because of the

seeming conflict. Suppose that 𝑔0 for all 𝑘. By

then 󰇛28󰇜, we get

𝑦

𝑑󰇛𝑔𝑔󰇜𝑑𝜇𝛼‖𝑑‖. (33)

In contrast, ‖𝑦‖‖𝑔𝑔‖𝐿‖𝑣‖.

Hence,

𝑔

𝑦‖𝑔‖‖𝑦‖𝐿‖𝑔‖‖𝑣‖. (34)

Therefore, from 󰇛21󰇜, we have

∥

𝑑

∥

∥𝑔󰇩𝛾𝛾𝐿∥

∥

𝑔∥

∥

𝑣∥

𝜇𝛼∥

∥

𝑑∥

∥



∥

𝑣



∥

𝑔

∥

𝛾𝛾𝐿∣𝑔 ∥

𝜇∣𝑑∥

∥

𝑣

∥

∥

𝑔∥

∥

𝛾𝛾𝐿𝜀𝛼

𝜇



√

𝜔∥𝑔 ∣

......󰇛35󰇜

where √𝜔𝛾𝛾

. This relation implies

that ∑



 

∥



∥



∑

 1∞. (36)

Therefore, 𝑙𝑖𝑚

→ 𝑖𝑛𝑓

∥

𝑔

∥

0.

4 Numerical Outcomes

This section presents the outcomes of the

computational experiments. A comparison has been

made between the standard HS algorithm and the

new scaled conjugate gradient method. Both

methods were tested using a Fortran

implementation. The test problems can be found in

references, [1], [16], [17], [18], [19]. In the

numerical experiments, the dimensions evaluated

were n=1000 and 10000 for each test function. We

selected fifteen large-scale, generalized, and

unconstrained problems. To determine the

termination of the process, we utilized the inequality

‖𝑔‖10 as the criterion. Table 1 provides a

view of the results derived from testing both

methods using the line parameters σ=0.001 and

𝛿0.9 in equations (4) and (5). The columns in

Table 1 indicate the following items

Problem: function name;

Dim: the dimension of the problem;

TNOI: total iterations number;

TIRS: total restarts number.

Table 1. Numerical comparison between the new

algorithm and HS Algorithm Comparison of

algorithms for 𝑛1000

Test Problems

New HS



NOI

IRS

NOI

IRS

Freudenstein and

Roth

137 125 838 810

Perturbed

Quadratic

328 94 392 116

Diagonal 2 199 56 200 64

Diagonal 3 1745 1591 F* F*

Extended Three

Expo Terms

28 21 25 18

Extended Powell 75 20 75 20

Extended

Maratos

64 29 85 44

Extended Cliff 11 9 13 10

Extended Wood 28 11 28 11

Quadratic QF2 395 122 403 121

EG2(CUTE) 113 48 F* F*

Tridiagonal

Perturbed Quad.

348 101 356 108

Drench (CUTE) 48 32 84 69

Diagonal 6 20 12 20 12

Sinquad (CUTE) 167 75 172 76

1848 707 2693 1479

Table 2. Numerical comparison between the

new algorithm and HS Algorithm for 𝑛

10000

Test Problems

New HS



NOI

IRS

NOI

IRS

Freudenstein and

Roth

15 8 F* F*

Perturbed

Quadratic

1223 326 1243 332

Diagonal 2 684 207 686 203

Diagonal 3 F* F* F* F*

Extended Three

Expo Terms

65 57 215 207

Extended Powell 82 26 97 29

Extended Maratos 64 29 64 28

Extended Cliff 16 11 24 22

Extended Wood 27 9 29 9

Quadratic QF2 1235 361 1285 368

EG2(CUTE) F* F* F* F*

Tridiagonal

Perturbed Quad.

1157 331 1297 376

Drench (CUTE) 51 40 102 89

Diagonal 6 18 9 22 11

Sinquad (CUTE) 437 29 1005 358

5059 1435 6069 2032

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.101

Isam H. Halil, Issam A.R. Moghrabi,

Ahmed A. Fawze, Basim A. Hassan, Hisham M. Khudhur

E-ISSN: 2224-2880

928

Volume 22, 2023

Table 2 above presents a numerical comparison

between the new algorithm and the HS Algorithm

for n=10000.

5 Conclusion

In this study, a novel method of scaled nonlinear

conjugate gradient is developed that, under certain

assumptions, boosts its global convergence property

for uniformly convex functions. Additionally, it

successfully fulfills the essential descent condition,

which is commonly observed in standard gradient

algorithms. The utility of the suggested new scaled

types was shown in the reported numerical

experiments. The obtained results reveal, for at least

the chosen set of test problems, that the developed

algorithm reduces by an overall 61% NOI and

78.82% IRS against the standard Hestenes-Stiefel

(HS) algorithm. The relative utility of the new

approach 󰇛𝑛1000,10000󰇜 is presented in Table

Table 3. Relative utility of the new approach 󰇛𝑛

1000,10000󰇜

Tools NOI IRS

HS-Algorithm 100 % 100%

New-Algorithm 39% 21.18 %

The new method is promising and deserves

further exploration of a wider spectrum of problems

by extending the method to constrained

optimization, exploring parallel and distributed

implementations for scalability, [8], [20], [21],

developing adaptive parameter tuning schemes, and

analyzing sensitivity to initial conditions. It is also

worth integrating the method with machine learning

models, [22], assessing robustness to noisy

objectives, comparing it with state-of-the-art

methods, exploring real-world applications, and

developing user-friendly interfaces for wider

accessibility. These directions aim to enhance the

algorithm's versatility, efficiency, and practical

applicability across diverse optimization scenarios.

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DOI: 10.37394/23206.2023.22.101

Isam H. Halil, Issam A.R. Moghrabi,

Ahmed A. Fawze, Basim A. Hassan, Hisham M. Khudhur

E-ISSN: 2224-2880

929

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

- Isam H. Halil contributed to the mathematical

derivations.

- Issam Moghrabi drafted the document and

contributed to the derivation.

- Ahmed Fawze carried out the numerical tests on

the new method.

- Basim Hassan did the coding necessary for

carrying out the tests.

- Hisham Khudhur summarized the results and did

proofreading.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.101

Isam H. Halil, Issam A.R. Moghrabi,

Ahmed A. Fawze, Basim A. Hassan, Hisham M. Khudhur

E-ISSN: 2224-2880

930

Volume 22, 2023