EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 1, Article Number 5551
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

New Newton Group Iterative Methods for Solving
Large-Scale Multi-Objective Constrained Optimization

Problems

Peng Cheng1,2, Jumat Sulaiman1,∗, Khadizah Ghazali1,
Majid Khan Majahar Ali3, Ming Ming Xu4

1 Faculty of Science and Natural Resources, Universiti Malaysia Sabah, Jalan UMS, 88400
Kota Kinabalu, Sabah, Malaysia
2 Chongqing College of Finance and Economics, 402160 Chongqing, China
3 School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Gelugor, Penang,
Malaysia
4 School of Mathematics and Information Technology, Xingtai University, 054000 Xingtai,
Hebei, China

Abstract. With the rapid development of big data and artificial intelligence technologies, we
are facing increasingly complex data and decision-making problems. Solving large-scale multi-
objective constrained optimization problems can help to solve many practical engineering and
scientific problems. The weighted and Lagrange multiplier methods are considered to be classical
and effective methods for dealing with multiple objectives and constraints, but there are some
difficulties in solving the processed unconstrained optimization problems. The Newton method is
a commonly used method for solving this type of problem, but it requires a high computational
complexity. In order to solve these difficulties, we combine four methods such as the Weighted
method, Lagrange multiplier method, Newton method and Explicit Group Gauss-Seidel iterative
method to propose new Newton Group iterative methods such as 2 and 4-point Explicit Group
Gauss-Seidel iterative methods namely as Newton-2EGGS and Newton-4EGGS for solving large-
scale multi-objective constrained optimization problems. Also, the convergence analysis of the
proposed method is presented. To test the superiority of the proposed method by comparing the
computational results, the Newton-4EGGS iteration is more efficient than both Newton-2EGGS
iterative method and the Newton-Gauss-Seidel iterative method (Newton-GS), especially in terms
of the number of iterations and the computational time.

2020 Mathematics Subject Classifications: 49M15, 65D15, 90C06, 90C29

Key Words and Phrases: Multi-objective constrained optimization, weighted method, Lagrange
multiplier method, Newton method, Explicit Group iterative method

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5551

Email addresses: pchengmath@163.com (P. Cheng),
jumat@ums.edu.my (J. Sulaiman), khadizah@ums.edu.my (K. Ghazali), majidkhanmajaharali@usm.my
(M. K. M. Ali), xmmzg@sina.com (M. M. Xu)

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



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 2 of 22

1. Introduction

With the development of modern science and technology and the progress of society,
many fields are faced with complex optimization problems, which often involve multi-
ple conflicting or interrelated objectives and are subject to multiple constraints at the
same time. Such problems are called multi-objective constrained optimization problems
(MOCOPs). Especially in the fields of industrial engineering, economic management, en-
vironmental protection, travelling salesman problem[37] and transportation [16, 36], the
emergence of multi-objective constrained optimization problems is increasingly frequent
and the scale is increasing, which puts higher demands on the efficiency and performance
of optimization algorithms. Traditional optimization algorithms are often incompetent
in dealing with such problems, while simple multi-objective optimization algorithms are
prone to fall into local optimum or computational inefficiency when facing large-scale prob-
lems [21, 25]. In addition, the existence of constraints further increases the complexity
of the problem, and making the algorithm more difficult to search for the optimal solu-
tion. Therefore, the study of effective algorithms for solving large-scale multi-objective
constrained optimization problems has important theoretical value and practical appli-
cation significance. It can not only promote the development of optimization algorithm
theory, but also provide strong technical support for solving practical engineering prob-
lems. The mathematical model of the multi-objective optimization problem (MOOP) with
constraints, which is mainly solved in this paper, can be expressed as follows:

min f(x) = (f1(x), f2(x), · · · , fn(x))T ,
s.t. gj(x) ≤ 0, j = 1, 2, · · · , p,

x ∈ Ω,

(1)

where, x = (x1, x2, · · · , xn)T , is the n-dimensional decision vector, Ω = [xLi , x
U
i ]

n ⊆
Rn, denotes the decision space, F (x) = (f1(x), f2(x), · · · , fn(x))T ∈ Rm is the objective
function and Rm is the m-dimensional objective space, gj(x) ≤ 0 is the j-th inequality
constraint, and let X denote the set of all feasible solutions to the problem (1), i.e.X =
{x ∈ Rn : gj(x) ≤ 0, j = 1, 2, · · · , p}.

The methods for solving multi-objective optimization problems primarily consist of
classical optimization techniques and intelligent optimization algorithms. Classical multi-
objective optimization methods encompass the linear weighted method, main objective
method, ideal point method [6], and evaluation function method [15]. Intelligent op-
timization algorithms mainly include Genetic Algorithms (GA) [22], Forbidden Search
Algorithm (Tabu Search) [18], Simulated Annealing Algorithm (SA) [3], Ant Colony Al-
gorithm (ACA) [30], Particle Swarm Optimization (PSO) algorithm [9] among others.
Intelligent optimization algorithms are a class of approaches that simulate the mechanism
of biological evolution in nature to conduct global probabilistic searches. These algorithms
do not require the search process to possess properties such as continuity, differentiability,
or convexity within the objective function space. They can obtain a set of nondomi-
nated solutions by decomposing a multi-objective problem into sub-problems to be solved.



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 3 of 22

However, these algorithms face challenges related to fitness allocation and diversity con-
trol as dimensionality increases. Additionally, due to their population-based nature, they
tend to have longer running times and are sensitive to parameter settings. On the other
hand, classical multi-objective optimization algorithms leverage characteristics of the so-
lution space such as differentiability and possess more well-established theories with lower
computational complexity and faster convergence speeds along with definite termination
criteria. Given that the linear weighted method in classical optimization techniques effec-
tively transforms multiple objective functions into a single one through linear weighted,
this paper primarily focuses on utilizing this approach for transforming problem (1).

Thus the problem (1) is converted into a solution on a single objective constrained
optimization problem (SOOP), where constraints play a crucial role by limiting the range
of feasible solutions and thus guiding the search process to find the optimal solution that
satisfies the actual problem. There are various methods for dealing with constraints, in-
cluding penalty function methods, Lagrange multiplier methods, projected gradient meth-
ods, feasible direction methods, genetic algorithms, evolutionary strategies, evolutionary
planning, and constrained variable scale methods [17, 23, 29, 32]. Among them, the La-
grange method simplifies the optimization problem by introducing Lagrange multipliers,
which transform the constraints of the original problem into the form of penalty terms.
Also the Lagrange method has higher flexibility in dealing with constraints compared to
traditional optimization algorithms. It can deal with a series of complex constraints, in-
cluding some nonlinear conditions, which has a great advantage in dealing with problems
with more complex constraints. Therefore, in this paper, the Lagrange method is used to
deal with the constraints of problem (1).

As for the solution of unconstrained optimization problems, the methods include: the
most rapid descent method [31], Newton method [20], the proposed Newton method [2],
etc., among which Newton method is the classical method for solving unconstrained opti-
mization problems, constructing Newton’s iterative direction under the condition that the
objective function and constraint function are second-order continuum differentiable, so
that the direction is a descending direction for each objective function under the condition
of feasibility, and then iteratively The optimal solution is finally obtained. Therefore, it
is very widely used, and this paper also mainly considers the Newton method to solve the
transformed unconstrained optimization problem.

In addition the Explicit Group iteration originally originated from the exploration of
numerical solutions of partial differential equations. Such equations have a wide range of
practical applications, including fluid dynamics, thermodynamics, electromagnetism, and
other fields. For such problems, it is often necessary to obtain their approximate solutions
through numerical computation. The traditional numerical solution methods, such as
the finite difference method, and the finite element method, although they can obtain
satisfactory solutions in many cases, often face challenges such as large computational
volume and poor stability when dealing with large-scale and high-complexity problems.
Therefore, researchers have begun to explore new numerical solution methods to address
these challenges. The group explicit iteration method is one of them. It was proposed
by Evans et al. in 1983 with the idea of group explicit (GE) [8]. Subsequently Saudi et



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 4 of 22

al. proposed an Explicit Group iteration based on the GE method suitable for robot path
planning [34, 35]. Sulaiman et al. [38] have proposed the concept of half-sweep and quarter-
sweep iterations respectively for solving their proposed problem. Since the application of
this method has been effectively promoted, Lung et al. [24] used the explicit group iteration
method to compute the solution of the telegraph equation, and Ghazali et al. combined this
method with Newton’s method to propose a method for solving unconstrained optimization
problems, which achieves computational efficiency and stability enhancement [12]. This
brings ideas to our research, and in this paper, we consider extending the Explicit Group
iteration into 2-point Explicit Group iteration and 4-point Explicit Group iteration, and
combining them with Gauss-Seidel iteration method for solving linear systems for solving
transformed unconstrained optimization problems.

Therefore, based on the research of Evans and Ghazali et al., we extend their works
for solving the (1) in this paper. The next part of this paper consists of the following main
parts: the second part introduces the weighted sum method and the Lagrange multiplier
method for transforming the problem (1) into an unconstrained single-objective optimiza-
tion problem, at the same time, we give the convergence proof; the third part presents
the new Newton Group iterative methods proposed in this paper. Numerical experiments
and presentation of results are given in the fourth part, and finally some conclusions are
discussed in the fifth part.

2. Formulation of weighted Lagrange method and its convergence
analysis

In this section, we give the main steps for transforming a multi-objective constrained
optimization problem (1) into a single-objective unconstrained optimization problem and
the relevant lemmas for establishing the equivalence of the corresponding problems. Based
on the equivalence lemmas, we give a convergence row analysis of the new method.

2.1. Weighted Lagrange Multiplier Method

We consider transforming the multi-objective constrained optimization problem into a
single-objective constrained optimization problem using the linear weighted sum method
with the following expression:

min F (x) = w1f1(x) + w2f2(x) + · · ·+ wnfn(x),

s.t. gj(x) ≤ 0, j = 1, 2, · · · , p,
x ∈ Ω.

(2)

where, wi ≤ 0(i = 1, . . . ,m) is the weight and
∑m

i=1 = 1.

Definition 1. [26] It is known that X ⊆ Rn is a feasible solution set for the model (1),
and obviously for (2), if X ⊆ X and there exists no x ∈ X such that

F (x) ≤ F (x∗)

then x∗ ∈ X is said to be an optimal solution to the constrained optimization problem (2).



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 5 of 22

The following lemma establishes the equivalence between the linear weighted problem
(2) and the solution of the original multi-objective optimization problem (1).

Lemma 1. [26] The optimal solution of the linear weighted problem (2) is a weak pareto
optimal solution of the multi-objective constrained optimization problem (1)). We denote
the weakly efficient set of (1) by X∗.

Lemma 2. [26] Let the multi-objective constrained optimization problem (1) be convex,
if x∗ is its pareto optimal solution, then there exists a weight vector w(wi > 0, i =
1, . . . ,m,

∑m
i=1 = 1 such that x∗ is the optimal solution of the weighted optimization prob-

lem (2).

In the following we transform (1) into the following unconstrained optimization prob-
lem by introducing a new variable λ, combining the inequality constraints as well as the
objective function F (x) using the Lagrange multiplier method, i.e.:

L(x, µ) = F (x) +

m∑
j=1

λjgj(x). (3)

where, F (x) = w1f1(x) + w2f2(x) + · · ·+ wnfn(x), λ = (λ1, λ2, · · · , λn)
T is the Lagrange

multiplier.

Lemma 3. [1]If x∗ is an optimal solution of the problem (3) and satisfies the following

KKT conditions, then x∗ is an optimal solution of the problem (2). We denote X̂∗
nas the

set of optimal solutions for (3).


▽xL(x

∗, λ) = ▽f(x∗) +
∑m

j=1 λjgj(x),

λjgj(x)
∗ = 0, j = 1, 2, · · · , p,

λj ≥ 0.

2.2. Convergence analysis of the proposed methods

In the following theorem, we prove that under certain conditions, the optimal solution
of the linear weighted problem (2) can be approximated by a convergent subsequence of the
sequence of optimal solutions of the unconstrained single-objective optimization problem
(3).

Theorem 1. Let x∗n be an optimal solution of (3), x∗nk
be a convergent subsequence of x∗n,

and lim
k→∞

x∗nk
= x∗ ∈ X∗, then the limit point of x∗nk

is an optimal solution of the linearly

weighted problem (2).

Proof. Since x∗nk
∈ x̂∗nk

then for all x ∈ X∗, we have

m∑
i=1

wifi(x
∗
nk
) +

p∑
j=1

λjgj(x
∗
nk
) ≤

m∑
i=1

fi(x) +

p∑
j=1

λjgj(x), k = 1, 2, · · · . (4)

We discuss this in the following two scenarios.



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 6 of 22

(i) If gj(x) = 0, it is clear that there are

lim
k→∞

p∑
j=1

λjgj(x
∗
nk
) = 0, (5)

lim
k→∞

p∑
j=1

λjgj(x) = 0, (6)

Taking the limit of equation (4) and combining the Equations (5) and (6)

lim
k→∞

m∑
i=1

wifi(x
∗
nk
) ≤ lim

k→∞

m∑
i=1

wifi(x),

i.e.
m∑
i=1

wifi(x
∗) ≤

m∑
i=1

wifi(x), for all x ∈ X∗.

So x∗ is the optimal solution of (2).

(ii) If gj(x) < 0, then there are

lim
k→∞

p∑
j=1

λjgj(x) ≤ 0, (7)

Since x∗n is an optimal solution of (3), and x∗nk
is a convergent subsequence of x∗n,

there are

lim
k→∞

p∑
j=1

λjgj(x
∗
nk
) = 0, (8)

Taking the limit of expression (3) and combining equations (7) and (8) yields

lim
k→∞

m∑
i=1

wifi(x
∗
nk
) ≤ lim

k→∞

m∑
i=1

wifi(x) + lim
k→∞

p∑
j=1

λjgj(x),

i.e.
m∑
i=1

wifi(x
∗) ≤

m∑
i=1

wifi(x),

So the theorem is proved.

From Theorem 1, Lemma 1, and Lemma 2, it can be obtained that the convergent
subsequence of the sequence of optimal solutions of the unconstrained single-objective
nonlinear programming problem (3) can approximate the weak pareto optimal solution of
the original multi-objective constrained optimization problem.



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 7 of 22

3. Newton-2EGGS and Newton-4EGGS

For the transformed problem (3), which is mainly a problem of unconstrained optimiza-
tion, combined with the advantages of Newton’s method of solving the problem they will
be used in the next solution, the iterative formula of Newton’s method can be expressed
as follows [4, 5, 13]:

x(k+1) = x(k) − αkd
(k),

d(k) = G−1
k gk, (9)

where, d(k) is the search direction, the step factor αk = 1, Gk = ▽2L(x(k)), gk = ▽L(x(k)),
and G−1

k is the inverse matrix of the Hessian matrix Gk = ▽2L(x(k)). Also, we know that
when Gk is positive definite, its inverse matrix must exist and be positive definite, so the
Newtonian direction will be a descent direction since it satisfies,

gTk d
(k) = −gTk G

−1
k gk < 0.

The Hessian matrix Gk is solved [14] as follows:

Gk =



∂2L/∂x21 ∂2L/(∂x1∂x2) · · · ∂2L/(∂x1∂xn) · · · ∂2L/(∂x1∂λp)
∂2L/(∂x2∂x1) ∂2L/∂x22 · · · ∂2L/(∂x2∂xn) · · · ∂2L/(∂x2∂λp)

...
...

. . .
...

. . .
...

∂2L/(∂xn∂x1) ∂2L/(∂xn∂x2) · · · ∂2L/∂x2n · · · ∂2L/(∂xn∂λp)
...

...
. . .

...
. . .

...
∂2L/(∂λp∂x1) ∂2L/(∂λp∂x2) · · · ∂2L/(∂λp∂xn) · · · ∂2L/∂λ2

p


,

The Newton equation (9) mentioned above, we can convert it into the following linear
system that

Gkdk = gk. (10)

After the k-th iteration, the specific coefficient matrix of the system of linear equations
(10) can be obtained by bringing xk into the Hessian matrix Gk. This coefficient matrix we
denote by the symbol A. Therefore, we write the system of equations (10) in the following
simple form:

Ad = b, (11)

where,

A =



a1,1 a1,2 · · · a1,n a1,n+1 · · · a1,s
a2,1 a2,2 · · · a2,n a2,n+1 · · · a2,s
...

...
. . .

...
...

. . .
...

an,1 an,2 · · · an,n an,n+1 · · · an,s
an+1,1 an+1,2 · · · an+1,n an+1,n+1 · · · an+1,s

...
...

. . .
...

...
. . .

...
as,1 as,2 · · · as,n as,n+1 · · · as,s


, d =


d1
d2
...

ds−1

ds

 , b =


b1
b2
...

bs−1

bs

 ,



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 8 of 22

a1,1, · · · , as,s, d1, · · · , ds, b1, · · · , bs ∈ R and s = n + p, the diagonal elements contain at
least p zeros. Combined with the Gauss-Seidel iterative method for solving the linear
system, we propose the 2-point Explicit Group Gauss-Seidel (2EGGS) iterative method
and the 4-point Explicit Group Gauss-Seidel (4EGGS) iterative method iteration. The
computational principles of the two methods are given below.

3.1. Formulation of Newton-2EGGS

To present our iterative method, we decompose the matrix, A as A = D1 − L1 − U1,
where D1, L1, U1 are of the following form:

D1 =



a1,1 a1,2 0 0 0 · · · 0 0
a2,1 a2,2 0 0 0 · · · 0 0
0 0 a3,3 a3,4 0 · · · 0 0
0 0 a4,3 a4,4 0 · · · 0 0

0 0 0 0
. . .

. . . 0 0
...

...
...

...
. . .

. . .
...

...
0 0 0 0 0 · · · as−1,s−1 as−1,s

0 0 0 0 0 · · · as,s−1 as,s


,

L1 =



0 0 a1,3 a1,4 a1,5 · · · a1,s−1 a1,s
0 0 a2,3 a2,4 a2,5 · · · a2,s−1 a2,s
0 0 0 0 a3,5 · · · a3,s−1 a3,s
0 0 0 0 a4,5 · · · a4,s−1 a4,s

0 0 0 0
. . .

. . . a5,s−1 a5,s
...

...
...

...
. . .

. . .
...

...
0 0 0 0 0 · · · 0 0
0 0 0 0 0 · · · 0 0


,

U1 =



0 0 0 0 0 · · · 0 0
0 0 0 0 0 · · · 0 0

a3,1 a3,2 0 0 0 · · · 0 0
a4,1 a4,2 0 0 0 · · · 0 0

a5,1 a5,2 a5,3 a5,4
. . .

. . . 0 0
...

...
...

...
. . .

. . .
...

...
as−1,1 as−1,2 as−1,3 as−1,4 as−1,5 · · · 0 0
as,1 as,2 as,3 as,4 as,5 · · · 0 0


,

If the last module of the block diagonal matrix, D1 is less than 4 elements, the actual
division of the elements will prevail. So the 2-point Explicit Group Gauss-Seidel iteration
for solving the linear system (11) is as follows:{

d(0)(initial vector),

d(k+1) = (D1 − L1)
−1U1d

(k) + (D1 − L1)
−1b, k = 0, 1, · · · .

(12)



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 9 of 22

The formula for calculating its components is given below and denoted as

d(k) =
(
d
(k)
1 , d

(k)
2 , · · · , d(k)i , · · · , d(k)s

)T
. (13)

From equation (12) we have

(D1 − L1)d
(k+1) = U1d

(k) + b,

D1d
(k+1) = L1d

(k) + U1d
(k) + b,

i.e.

ai,id
(k+1)
i + ai,i+1d

(k+1)
i+1 = −

i−1∑
j=1

ai,jd
(k+1)
j −

s∑
j=i+2

ai,jd
(k)
j + bi, i = 1, 2, · · · , s. (14)

ai+1,id
(k+1)
i + ai+1,i+1d

(k+1)
i+1 = −

i−1∑
j=1

ai+1,jd
(k+1)
j −

s∑
j=i+2

ai+1,jd
(k)
j + bi+1, i = 1, 2, · · · , s.

(15)

Solving equations (14) and (15) yields the following formula for solving the 2-point Explicit
Group Gauss-Seidel iterative method component of the linear system (11).

d(0) =
(
d
(0)
1 , d

(0)
2 , · · · , d(0)i , · · · , d(0)s

)T
,

d
(k+1)
i =

ai+1,i+1Q1−ai,i+1Q2

ω ,

d
(k+1)
i+1 =

−ai+1,iQ1+ai,iQ2

ω ,

Q1 = −
∑i−1

j=1 ai,jd
(k+1)
j −

∑s
j=i+2 ai,jd

(k)
j + bi,

Q2 = −
∑i−1

j=1 ai+1,jd
(k+1)
j −

∑s
j=i+2 ai+1,jd

(k)
j + bi+1,

ω = ai,iai+1,i+1 − ai,i+1ai+1,i,

i = 1, 2, · · · , s; k = 0, 1, · · · .

(16)

So we get the iteration steps of the Newton-2EGGS iterative method as follows:
Algorithm 1: Newton-2EGGS Scheme

I. Calculate the expression of the objective function by using equation (3).

II. Give the initial value x0 and the accuracy threshold ε1 = 10−5, ε2 = 10−10 and let
k := 0.

III. Calculate matrix Gk and gradient gk If ∥gk∥ < ε1, that is, the extreme point is
reached, the iteration is stopped, otherwise, go to IV.

IV. Calculate the following



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 10 of 22

a. Calculate the matrix D1,L1 and U1.

b. Calculate the search direction by using equation (16).

c. Calculate the convergence condition, if ∥d(k+1) − d(k)∥ < ε2, then go to step V,
otherwise go to step IV(b).

V. Calculate the new iteration point as x(k+1) = x(k) + αkd
(k).

VI. Let k = k + 1, go to step III.

3.2. Formulation of Newton-4EGGS

Similar to the 2EGGS iterative method, we decompose the matrix, A as A = D2 −
L2 − U2, where

D2 =



V1 0 · · · 0 · · · 0
0 V2 · · · 0 · · · 0
...

...
. . .

...
. . .

...
0 0 · · · Vi · · · 0
...

...
. . .

...
. . .

...
0 0 · · · 0 · · · Vq


, L2 = −



0 0 · · · 0 · · · 0
W2,1 0 · · · 0 · · · 0
...

...
. . .

...
. . .

...
Wi,1 Wi,2 · · · 0 · · · 0
...

...
. . .

...
. . .

...
Wq,1 Wq,2 · · · Wq,i · · · 0


,

U2 = −



0 W1,2 · · · W1,i · · · W1,q

0 0 · · · W2,i · · · W2,q
...

...
. . .

...
. . .

...
0 0 · · · 0 · · · Wi,q
...

...
. . .

...
. . .

...
0 0 · · · 0 · · · 0


,

Vi,i =


ai,i ai,i+1 ai,i+2 ai,i+3

ai+1,i ai+1,i+1 ai+1,i+2 ai+1,i+3

ai+2,i ai+2,i+1 ai+2,i+2 ai+2,i+3

ai+3,i ai+3,i+1 ai+3,i+2 ai+3,i+3

 ,Wi,j =


ai,s−3 ai,s−2 ai,s−1 ai,s
ai+1,s−3 ai+1,s−2 ai+1,s−1 ai+1,s

ai+2,s−3 ai+2,s−2 ai+2,s−1 ai+2,s

ai+3,s−3 ai+3,s−2 ai+3,s−1 ai+3,s

 .

If there are less than 16 elements inside the final modular matrix Vq, then the actual
possession will prevail. So the 4-point Explicit Group Gauss-Seidel iterative method for
solving the linear system (11) is as follows.{

d(0)(initial vector),

d(k+1) = (D2 − L2)
−1U2d

(k) + (D2 − L2)
−1b, k = 0, 1, · · · .

(17)

The formula for calculating its components is (13). From equation (17) we have

D2d
(k+1) = L2d

(k) + U2d
(k) + b,



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 11 of 22

i.e.

i+3∑
j=i

ai,jd
(k+1)
j = −

i−1∑
j=1

ai,jd
(k+1)
j −

s∑
j=i+4

ai,jd
(k+1)
j + bi, (18)

i+3∑
j=i

ai+1,jd
(k+1)
j = −

i−1∑
j=1

ai+1,jd
(k+1)
j −

s∑
j=i+4

ai+1,jd
(k+1)
j + bi+1, (19)

i+3∑
j=i

ai+2,jd
(k+1)
j = −

i−1∑
j=1

ai+2,jd
(k+1)
j −

s∑
j=i+4

ai+2,jd
(k+1)
j + bi+2, (20)

i+3∑
j=i

ai+3,jd
(k+1)
j = −

i−1∑
j=1

ai+3,jd
(k+1)
j −

s∑
j=i+4

ai+3,jd
(k+1)
j + bi+3, (21)

Solving equations (18)-(21) yields

d(0) =
(
d
(0)
1 , d

(0)
2 , · · · , d(0)i , · · · , d(0)s

)T
,

d
(k+1)
i = B1

|Vi| ,

d
(k+1)
i+1 = B2

|Vi| ,

d
(k+1)
i+2 = B3

|Vi| ,

d
(k+1)
i+3 = B4

|Vi| ,

i = 1, 2, · · · , s; k = 0, 1, · · · .

(22)

where,

B1 =

∣∣∣∣∣∣∣∣
Qi ai,i+1 ai,i+2 ai,i+3

Qi+1 ai+1,i+1 ai+1,i+2 ai+1,i+3

Qi+2 ai+2,i+1 ai+2,i+2 ai+2,i+3

Qi+3 ai+3,i+1 ai+3,i+2 ai+3,i+3

∣∣∣∣∣∣∣∣ , B2 =

∣∣∣∣∣∣∣∣
ai,i Qi ai,i+2 ai,i+3

ai+1,i Qi+1 ai+1,i+2 ai+1,i+3

ai+2,i Qi+2 ai+2,i+2 ai+2,i+3

ai+3,i Qi+3 ai+3,i+2 ai+3,i+3

∣∣∣∣∣∣∣∣ ,

B3 =

∣∣∣∣∣∣∣∣
ai,i ai,i+1 Qi ai,i+3

ai+1,i ai+1,i+1 Qi+1 ai+1,i+3

ai+2,i ai+2,i+1 Qi+2 ai+2,i+3

ai+3,i ai+3,i+1 Qi+3 ai+3,i+3

∣∣∣∣∣∣∣∣ , B4 =

∣∣∣∣∣∣∣∣
ai,i ai,i+1 ai,i+2 Qi

ai+1,i ai+1,i+1 ai+1,i+2 Qi+1

ai+2,i ai+2,i+1 ai+2,i+2 Qi+2

ai+3,i ai+3,i+1 ai+3,i+2 Qi+3

∣∣∣∣∣∣∣∣ ,
i.e.

Qi = −
i−1∑
j=1

ai,jd
(k+1)
j −

s∑
j=i+4

ai,jd
(k+1)
j + bi,



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 12 of 22

Qi+1 = −
i−1∑
j=1

ai+1,jd
(k+1)
j −

s∑
j=i+4

ai+1,jd
(k+1)
j + bi+1,

Qi+2 = −
i−1∑
j=1

ai+2,jd
(k+1)
j −

s∑
j=i+4

ai+2,jd
(k+1)
j + bi+2,

Qi+3 = −
i−1∑
j=1

ai+3,jd
(k+1)
j −

s∑
j=i+4

ai+3,jd
(k+1)
j + bi+3,

Algorithm 2: Newton-4EGGS Scheme

I. Calculate the expression of the objective function by using equation (3).

II. Give the initial value x0 and the accuracy threshold ε1 = 10−5, ε2 = 10−10 and let
k := 0.

III. Calculate matrix Gk and gradient gk If ∥gk∥ < ε1, that is, the extreme point is
reached, the iteration is stopped, otherwise, go to IV.

IV. Calculate the following

a. Calculate the matrix D2,L2,U2,Vi and Wi,q.

b. Calculate the search direction by using equation (22).

c. Calculate the convergence condition, if ∥d(k+1) − d(k)∥ < ε2, then go to step V,
otherwise go to step IV(b).

V. Calculate the new iteration point as x(k+1) = x(k) + αkd
(k).

VI. Let k = k + 1, go to step III.

4. Numerical experiments

In this section, numerical experiments are performed to verify the effectiveness and
superiority of the algorithm, the test functions are mainly taken from references [10, 11, 19].
They are all chosen based on the fact that the Hessian matrix is a full matrix. These test
cases were all computed at n = (10, 50, 100, 150, 150) for a total of five different orders
of Hessian matrices. Thus, this corresponds to a total of 20 test cases provided. All
calculations and drawings in this paper are done by using MATLAB software, version:
MATLAB2020a. Below we give the specific expressions for the four cases.

Example 1[10]

min f1(x) = x1 + g,



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 13 of 22

min f2(x) = 1− x21 + g,

g =
n∑

i=2

(xi − sin(0.5πx1))
2,

s.t. c(x) = sin(aπx1)− b ≥ 0, x ∈ [0, 1],

where a > 0, b ∈ [−1, 1]. As an example, a = 1, b = 0.5 are set here.
Firstly, the Example1 is transformed into a single objective optimization problem using

weighted as follows:

min F (x) = w1f1(x) + w2f2(x),

s.t. c(x) = sin(aπx1)− b ≥ 0,

x ∈ Ω.

where, w is the weight coefficient, and
∑2

i=1wi = 1. The Lagrange multiplier method is
then used to transform Example 1 into an unconstrained optimization problem as follows:

min F (x) = w1f1(x) + w2f2(x) + λ(−c(x)),

where, λ > 0 is the Lagrange multiplier. This example has an external Newton method
iteration number of 3 for dimensions 10, 50, and 4 for dimensions 100, 150, and 250 for
an initial point of (1, 1, ..., 1), and the objective function values are all approximated as
f1(x) = 0.8333, f2(x) = 0.3056. Figure 1 illustrates a three-dimensional stereogram of
the weighted Lagrange function F1(x) for dimension n = 2. The following examples are
converted using a similar method as in Example 1.

Example 2[11]

min f1(x) = x1 + g1,

min f2(x) = 1− x21 + g2,

g1 =
∑
i∈I1

[xi − sin(0.5πx1)]
2 ,

g2 =
∑
i∈I2

[xi − cos(0.5πx1)]
2 ,

s.t. c(x) = sin(aπx1)− b ≥ 0,

I1 = {i|i is odd and 2 ≤ i ≤ n} ,
I2 = {i|i is even and 2 ≤ i ≤ n} ,

x ∈ [0, 1],

where a > 0, b ∈ [−1, 1]. As an example, a = 2, b = 0.1 are set here.
The weighted Lagrange function for Example 2 is

min F2(x) = w1(x1 + g1) + w2(1− x21 + g2) + λ(−sin(2πx1)− 0.1).



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 14 of 22

This example has an external Newton method iteration number of 2 for dimensions 10,
50, and 3 for dimensions 100, 150, and 250 for an initial point of (0.5,0.4,0.5, 0.4,. . . ,0.5,0.4),
and the objective function values are all approximated as f1(x) = 0.4841, f2(x) = 0.7657.
Figure 2 illustrates a three-dimensional stereogram of the weighted Lagrange function
F2(x) for dimension n = 2.

Example 3[11]

min f1(x) = x1 + g1,

min f2(x) = 1−
√
x1 + g2,

g1 =
∑
i∈I1

[xi − sin(0.5πx1)]
2 ,

g2 =
∑
i∈I2

[xi − cos(0.5πx1)]
2 ,

s.t. c(x) = sin(aπx1)− b ≥ 0,

I1 = {i|i is odd and 2 ≤ i ≤ n} ,
I2 = {i|i is even and 2 ≤ i ≤ n} ,

x ∈ [0, 1],

where a > 0, b ∈ [−1, 1]. As an example, a = 2, b = 0.2 are set here.
The weighted Lagrange function for Example 2 is

min F3(x) = w1(x1 + g1) + w2(1−
√
x1 + g2) + λ(−sin(2πx1)− 0.2).

Example 3 has an external Newton method iteration number of 2 for dimensions 10 and
3 for dimensions 50, 100, 150, and 250 for an initial point of (0.5,0.8,0.5,0.8,. . . , 0.5,0.8),
and the objective function values are all approximated as f1(x) = 0.4680,f2(x) = 0.3159.
Figure 3 illustrates a three-dimensional stereogram of the weighted Lagrange function
F3(x) for dimension n=2.

Example 4[19]

min f1(x) = (1 + g)cos(
x1π

2
) · · · cos(xM−1π

2
),

min f1(x) = (1 + g)cos(
x1π

2
) · · · sin(xM−1π

2
),

...

min fM (x) = (1 + g)sin(
x1π

2
),

g =
n∑

i=M

(xi − 0.5)2,

s.t. c(x) =
M∑
i=1

(fi(x)− σ)2 − r2 ≥ 0, x ∈ [0, 1],



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 15 of 22

where M = 2, σ = 1
M

∑M
i=1 fi(x), r = 0.225. The weighted Lagrange function for Example

2 is

min F2(x) = w1f1(x) + w2f2(x)

+ λ

{
−

[(
f1(x)−

1

2
(f1(x) + f2(x))

)2

+

(
f2(x)−

1

2
(f1(x) + f2(x))

)2

− 0.2252

]}
,

where, f1(x) = (1 + g)cos(x1π
2 ), f2(x) = (1 + g)sin(x1π

2 ), g(x) = (x2 − 0.5)2.
This example has 4 external Newton’s method iterations for dimension 10, 5 external

for dimension 50, 6 external for dimensions 100 and 150, and 8 external for dimension
250 with initial points (0.2,0.4,0.3,0.4,...,0.3,0.4), and the objective function values are all
approximated by f1(x) = 0.9579, f2(x) = 0.2871. Figure 4 illustrates a three-dimensional
stereogram of the weighted Lagrangian function F4(x) for dimension n = 2. The three-
dimensional stereograms of Figures 1-Figure 4 show the location of the minimum value
of the weighted Lagrange function in the feasible domain for the four examples. The
explanation of the symbols is first given by Table 1.

Table 1: Description of symbols used in the result display.

Notation Description

n Test number
N Number of variables
GS Gauss-Seidel method

2EGGS 2-point Explicit Group GS
4EGGS 4-point Explicit Group GS
NGS Newton-GS method

2NEGS Newton-2EGGS method
4NEGS Newton-4EGGS method
TM Computational time(Unit: Second)
NOI Internal iterations
L2− g The L2 Norm of the gradient g at the end of the computation

Figure 1: 3D diagram for Example 1. Figure 2: 3D diagram for Example 2.



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 16 of 22

Figure 3: 3D diagram for Example 3. Figure 4: 3D diagram for Example 4.

Table 2: Comparison of the results of the above three different methods for Example 1.

N
P method 10 50 100 150 250

NGS 105 903 3788 8111 21055
NOI 2NEGGS 100 893 3761 8071 20992

4NEGGS 91 872 3709 7994 20867

NGS 0.8978 10.2485 79.5447 291.4592 1785.2087
TM 2NEGGS 0.8658 9.3863 75.8597 290.8716 1757.3603

4NEGGS 0.8136 9.1721 74.8187 280.9366 1718.8262

NGS 8.9251E-07 5.3210E-06 5.0076E-09 9.2859E-09 1.9899E-08
L2-g 2NEGGS 8.9251E-07 5.3210E-06 5.0072E-09 9.2687E-09 1.9836E-08

4NEGGS 8.9251E-07 5.3210E-06 4.8564E-09 9.1065E-09 1.9702E-08

Figure 5: iterations and time (seconds) difference in different dimensions for Example 1.

5. Comparison Results

We show specific computational results for four different example problems in Tables 2-
5, which indicate that our proposed new explicit group iteration method successfully solves
large-scale multi-objective constrained optimization problems. Since the same pareto op-
timal solution is obtained using different iteration methods for the same latitude case in
each example problem. So we only show the number of iterations inside the algorithm, the



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 17 of 22

Table 3: Comparison of the results of the above three different methods for Example 2.

N
P method 10 50 100 150 250

NGS 272 2679 7140 11425 23228
NOI 2NEGGS 262 2577 7110 11398 23203

4NEGGS 260 2534 7040 11333 23129

NGS 1.4114 20.1857 118.9396 352.2824 1790.0120
TM 2NEGGS 1.3655 19.5404 115.9887 349.8387 1779.2555

4NEGGS 1.3561 19.2925 114.1318 345.3602 1763.0075

NGS 9.5525E-07 5.3173E-06 7.2858E-09 1.8530E-08 4.4736E-08
L2-g 2NEGGS 9.5525E-07 5.3173E-06 7.3544E-09 1.8463E-08 4.4585E-08

4NEGGS 9.5525E-07 5.3174E-06 7.5201E-09 1.8393E-08 4.4297E-08

Figure 6: iterations and time (seconds) difference in different dimensions for Example 2.

Table 4: Comparison of the results of the above three different methods for Example 3.

N
P method 10 50 100 150 250

NGS 323 3615 8069 12715 26828
NOI 2NEGGS 305 3306 8015 12676 26755

4NEGGS 280 3258 7914 12594 26587

NGS 1.6175 27.9083 130.9371 389.1743 2031.1232
TM 2NEGGS 1.5249 25.3495 128.6421 382.9091 2029.2933

4NEGGS 1.4355 25.0829 125.1750 380.2002 1995.1464

NGS 6.2168E-06 3.2971E-09 8.9170E-09 1.3044E-08 4.0726E-08
L2-g 2NEGGS 6.2168E-06 4.9737E-09 8.7289E-09 1.3035E-08 4.0622E-08

4NEGGS 6.2168E-06 4.4817E-09 8.3910E-09 1.3006E-08 4.0363E-08

total computation time and the Norm of Function Gradient at Termination of Calculation
inside the table. For comparison purposes we keep the decimals in Tables 2-5 to 4 decimal
places. In addition, Figure 5-8 shows the difference between the number of iterations and
the computation time between different algorithms for each example problem in different



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 18 of 22

Figure 7: iterations and time (seconds) difference in different dimensions for Example 3.

Table 5: Comparison of the results of the above three different methods for Example 4.

N
P method 10 50 100 150 250

NGS 91 177 1281 7269 114176
NOI 2NEGGS 91 172 1167 6234 86083

4NEGGS 90 157 913 4267 48913

NGS 1.1596 11.1179 69.6460 352.2824 8487.6144
TM 2NEGGS 1.1349 10.1160 64.7557 339.1660 6584.3840

4NEGGS 1.0560 10.0277 62.9561 305.9793 4115.1276

NGS 1.7766E-07 4.3005E-08 2.4160E-06 8.9724E-06 1.8916E-06
L2-g 2NEGGS 1.7766E-07 4.3005E-08 2.4160E-06 8.9724E-06 1.8916E-06

4NEGGS 1.7766E-07 4.3005E-08 2.4160E-06 8.9724E-06 1.8916E-06

Figure 8: iterations and time (seconds) difference in different dimensions for Example 4.

dimensions (NGS-4NEGGS denotes the result of the Newton-GS method minus the com-
putation result of Newton-4EGGS), so the clear advantage of Newton-4EGGS can be seen
from both Tables 2-5 and Figures 5-8.

From the results in Table 6, it can be seen that the number of iterations using the
Newton-2EGGS method is less than that using the Newton-GS method and the percentage
reduction can be up to 24.60. And here the Newton-4EGGS method is the most compu-
tationally efficient and the reduction in the number of iterations compared to Newton-GS



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 19 of 22

Table 6: Decreasing Percentage of Iterations of Newton-4EGGS Relative to Newton-GS and Newton-2EGGS

Range of decreasing percentage of iterations
n NGS − 2NEGGS NGS − 4NEGGS 2NEGGS − 4NEGGS

1 0.3 ∽ 4.76 0.89 ∽ 13.33 0.60 ∽ 9.00
2 0.11 ∽ 3.81 0.43 ∽ 5.41 0.32 ∽ 1.67
3 0.27 ∽ 8.55 0.90 ∽ 13.31 0.63 ∽ 8.20
4 0.00 ∽ 24.6 1.11 ∽ 57.16 1.11 ∽ 43.18

Table 7: Decreasing Percentage of computational time of Newton-4EGGS Relative to Newton-GS and Newton-
2EGGS

Computing time

n NGS(I) 2NEGGS(II) 4NEGGS(III) (I−II)
I

(I−III)
I

(II−III)
II

1 2167.36 2134.34 2084.57 1.52 3.82 2.33
2 2282.83 2265.99 2243.15 0.74 1.74 1.01
3 2580.76 2567.72 2527.04 0.51 2.08 1.58
4 8908.70 6966.37 4450.22 21.8 50.05 36.12

and Newton-2EGGS can be up to 54.16% and 43.18%.
As expected, the two methods we used are much faster to compute than the reference

method. Also in Table 7, it can be seen that the computation time of our proposed
Newton-4EGGS is up to 50.05% faster than the Newton-GS method, and the Newton-
4EGGS method is in turn 36.12% faster than the Newton-2EGGS, so it can be concluded
that our proposed iterative method is able to achieve a substantial improvement in the
number of iterations and the execution time over the reference Newton-GS method and
Newton-4EGGS performs better.

6. Conclusion

This paper focuses on algorithms for solving large-scale multi-objective constrained
optimization problems. Based on the characteristics of the objective functions and con-
straints, new Newton Group iterative methods is proposed based on the classical optimiza-
tion algorithms: weighting, Lagrange multiplier, and Newton method. The convergence
of the method is also theoretically analyzed, and then the method is used in four different
examples, and the computational results of the new method and the reference method in
terms of the number of iterations and the computational time are shown in the form of
tables and graphs. Finally, a comparative analysis verifies the obvious superiority of our
proposed method compared with the reference Newton-Gauss-Seidel method, especially
in terms of the number of iterations and computational time, which greatly reduces the
computational complexity. This provides a new idea for solving large-scale multi-objective
constrained optimization problems. Then how to innovate the method to make the com-
putation more efficient or to compute the ultra-large scale multi-objective optimization
problem will be the direction of our next research. For future work, this study will be



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 20 of 22

extended to investigate on the use of two-step iterative methods such as arithmetic mean
[27, 33], Geometric mean[28] and AGE[7] as a smoother combined with the Newton method
to solve the proposed multi-objective constrained optimization problems.

Acknowledgements

The research was found by the Universiti Malaysia Sabah’s UMSGreat Research Grant
No. GUG0568-1/2022.

References

[1] M S Bazaraa, H D Sherali, and C M Shetty. Nonlinear multiobjective optimization.
John wiley & sons, New York, USA, 2006.

[2] R H Byrd and J. Nocedal. A tool for the analysis of quasi-Newton methods with
application to unconstrained minimization. SIAM Journal on Numerical Analysis,
26(3):727–739, 1989.

[3] B Cakir, F Altiparmak, and B Dengiz. Multi-objective optimization of a stochastic
assembly line balancing: A hybrid simulated annealing algorithm. Computers &
Industrial Engineering, 60(3):376–384, 2011.

[4] P Cheng, J Sulaiman, K Ghazali, M K M Ali, and M M Xu. Application of newton-
sor iteration with linear weighted lagrange approach for solving multi-objective con-
strained optimization problems. In In International Conference on Advances in Com-
putational Science and Engineering, pages 3–16, Manila, Philippines, 2024. Springer
Nature Singapore.

[5] P Cheng, J Sulaiman, K Ghazali, M K M Ali, and M M Xu. Newton-SOR Iterative
Method with Lagrangian Function for Large-Scale Nonlinear Constrained Optimiza-
tion Problems. Journal of Advanced Research in Applied Sciences and Engineering
Technology, 46(2):251–262, 2025.

[6] K Deb, K Sindhya, and J Hakanen. Multi-objective optimization. Decision sciences,
CRC Press, 2016.

[7] D J Evans. The alternating group explicit (AGE) matrix iterative method. Applied
Mathematical Modelling, 11(4):256–263, 1987.

[8] D J Evans and A R B Abdullah. Group explicit methods for parabolic equations.
International journal of computer mathematics, 14(1):73–105, 1983.

[9] S K S Fan and J M Chang. A parallel particle swarm optimization algorithm for multi-
objective optimization problems. Engineering Optimization, 41(7):673–697, 2009.

[10] Z Fan, W Li X Cai, H Li C Wei, Q Zhang, K Deb, and E Goodman. Difficulty
adjustable and scalable constrained multiobjective test problem toolkit. Evolutionary
computation, 28(3):339–378, 2020.

[11] Z Fan, W Li, X Cai, H Li, K Hu, and H Yin. Difficulty controllable and scal-
able constrained multi-objective test problems. In 2015 International Conference
on Industrial Informatics-Computing Technology, Intelligent Technology, Industrial
Information Integration., pages 76–83, Wuhan, China, 2015. IEEE.



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 21 of 22

[12] K Ghazali, J Sulaiman, Y Dasril, and D Gabda. Newton method with explicit group
iteration for solving large scale unconstrained optimization problems. In Journal
of Physics: Conference Series., volume 1132. IOP Publishing, Putrajaya, Malaysia,
2018.

[13] K Ghazali, J Sulaiman, Y Dasril, and D Gabda. Newton-MSOR Method for Solv-
ing Large-Scale Unconstrained Optimization Problems with an Arrowhead Hessian
Matrices. Transactions on Science and Technology, 6(2-2):228–234, 2019.

[14] K Ghazali, J Sulaiman, Y Dasril, and D Gabda. An improvement of computing
newton’s direction for finding unconstrained minimizer for large-scale problems with
an arrowhead hessian matrix. In Computational Science and Technology: 6th ICCST
2019., volume 1132. Springer Singapore, Kota Kinabalu, Malaysia, 2020.

[15] N Gunantara. A review of multi-objective optimization: Methods and its applications.
Cogent Engineering, 5(1):1–16, 2018.

[16] Y Y Han, D W Gong, X Y Sun, and Q K Pan. An improved NSGA-II algorithm for
multi-objective lot-streaming flow shop scheduling problem. International Journal of
Production Research, 52(8):2211–2231, 2014.

[17] M Hassan and A Baharum. A new logarithmic penalty function approach for nonlinear
constrained optimization problem. Decision Science Letters, 8(3):353–362, 2019.

[18] D M Jaeggi, G T Parks, T Kipouros, and P J Clarkson. The development of a multi-
objective Tabu Search algorithm for continuous optimisation problems. European
Journal of Operational Research, 185(3):1192–1212, 2008.

[19] H Jain and K Deb. An evolutionary many-objective optimization algorithm using
reference-point based nondominated sorting approach, part II: Handling constraints
and extending to an adaptive approach. IEEE Transactions on evolutionary compu-
tation, 18(4):602–622, 2013.

[20] J E Dennis Jr and R B Schnabel. Numerical methods for unconstrained optimization
and nonlinear equations. Society for Industrial and Applied Mathematics, USA, 1996.

[21] J R Kasprzyk, P M Reed, B R Kirsch, and G W Characklis. Managing population
and drought risks using many-objective water portfolio planning under uncertainty.
Water Resources Research, 45(12):1–18, 2009.

[22] A Konak, D W Coit, and A E Smith. Multi-objective optimization using genetic
algorithms: A tutorial. Reliability engineering & system safety, 91(9):992–10078,
2006.

[23] Y P Laptin and T O Bardadym. Problems related to estimating the coefficients of
exact penalty functions. Cybernetics and systems analysis, 55:400–412, 2019.

[24] J V Lung and J Sulaiman. On quarter-sweep finite difference scheme for one-
dimensional porous medium equations. International Journal of Applied Mathematics,
33(3):439, 2020.

[25] R J Lygoe, M Cary, and P J Fleming. A many-objective optimisation decision-making
process applied to automotive diesel engine calibration. In K Deb, editor, Simulated
Evolution and Learning., volume 11b. Springer, Kanpur, India, 2010.

[26] K Miettinen. Nonlinear multiobjective optimization. Springer Science & Business
Media, New York, USA, 1999.



P. Cheng et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5551 22 of 22

[27] M S Muthuvalu and J Sulaiman. Quarter-Sweep Arithmetic Mean (QSAM) iterative
method for second kind linear Fredholm integral equations. Appl. Math. Sci, 4:2943–
2953, 2010.

[28] M Sundaram Muthuvalu and J Sulaiman. Half-sweep geometric mean iterative
method for the repeated Simpson solution of second kind linear Fredholm integral
equations. Proyecciones (Antofagasta), 31(1):65–79, 2012.

[29] V H Nguyen and J J Strodiot. On the convergence rate for a penalty function method
of exponential type. Journal of Optimization Theory and Applications, 27:495–508,
1979.

[30] J Ning, C Zhang, P Sun, and Y Feng. Comparative study of ant colony algorithms
for multi-objective optimization. Information, 10(1):11, 2018.

[31] J Nocedal. Theory of algorithms for unconstrained optimization. Acta numerica,
1:199–242, 1992.

[32] R T Rockafellar. The multiplier method of Hestenes and Powell applied to convex
programming. Journal of Optimization Theory and applications, 12(6):555–562, 1973.

[33] V Ruggiero and E Galligani. An iterative method for large sparse linear systems on
a vector computer. Computers & Mathematics with Applications, 20(1):25–28, 1990.

[34] A Saudi and J Sulaiman. Red-black strategy for mobile robot path planning. In
Proceedings of the international multiconference of engineers and computer scientists,
Hong Kong, China, 2010. International Association of Engineers.

[35] A Saudi and J Sulaiman. Robot path planning using four point-explicit group via
nine-point laplacian (4EG9L) iterative method. Procedia Engineering, 41:182–188,
2012.

[36] M. K. Sharma, A. K. Bhargava, S. Kumar, L. Rathour, L. N. Mishra, and S. Pandey. A
fermatean fuzzy ranking function in optimization of intuitionistic fuzzy transportation
problems. Advanced Mathematical Models and Applications, 7(2):191–204, 2022.

[37] M K Sharma, S Chaudhary, L Rathour, and V N Mishra. Modified genetic algorithm
with novel crossover and mutation operator in travelling salesman problem. Sigma
Journal of Engineering and Natural Sciences, 42(6), 2024.

[38] J Sulaiman, M Othman, and M K Hasan. Red-black half-sweep iterative method using
triangle finite element approximation for 2d poisson equations. In Y Shi, G D Al-
bada, J Dongarra, and P M A Sloot, editors, Computational Science–ICCS 2007: 7th
International Conference., volume 4487. Springer Berlin Heidelberg, Beijing, China,
2007.