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Solving systems of nonlinear equations is a relatively complicated problem in which arise in a diverse range of sciences. There are a number of different approaches have been proposed. In this paper, we employ the imperialist competitive algorithm (ICA) for solving systems of nonlinear equations. Some well-known problems are presented to demonstrate the efficiency of this new robust optimization method in comparison to other known methods.
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Computers and Mathematics with Applications 65 (2013) 1894–1908
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Computers and Mathematics with Applications
journal homepage: www.elsevier.com/locate/camwa
Imperialist competitive algorithm for solving systems of
nonlinear equations
Mahdi Abdollahi a,, Ayaz Isazadeh b, Davoud Abdollahi c
aUniversity of Tabriz, Aras International Campus, Department of Computer Sciences, P. O. Box 51666-16471, Islamic Republic of Iran
bUniversity of Tabriz, Department of Computer Sciences, Islamic Republic of Iran
cUniversity College of Daneshvaran, Tabriz, Islamic Republic of Iran
article info
Article history:
Received 3 September 2012
Received in revised form 29 January 2013
Accepted 6 April 2013
Keywords:
ICA
Nonlinear equations
Root solvers
Evolutionary multi-objective optimization
Meta-heuristics
abstract
Solving systems of nonlinear equations is a relatively complicated problem in which arise
a diverse range of sciences. There are a number of different approaches that have been
proposed. In this paper, we employ the imperialist competitive algorithm (ICA) for solving
systems of nonlinear equations. Some well-known problems are presented to demonstrate
the efficiency of this new robust optimization method in comparison to other known
methods.
©2013 Elsevier Ltd. All rights reserved.
1. Introduction
Systems of nonlinear equations arise in a diverse range of sciences such as economics, engineering, chemistry, mechanics,
medicine and robotics. The problem is nondeterministic polynomial-time hard when the equations in the system do not
exhibit nice linear or polynomial properties. However, a number of different approaches have been proposed such as Luo
et al. [1] and Mo et al. [2] used a combination of chaos search and Newton type methods and a combination of the conjugate
direction method (CD) respectively. In the same way, M. Jaberipour [3] used particle swarm algorithm but there still exist
some obstacles in solving systems of nonlinear equations. The most widely used algorithms are Newton-type methods,
though their convergence and effective performance can be highly sensitive to the initial guess of the solution supplied to
the methods. So the algorithm would fail with the improper initial guess. For this reason, it is necessary to find an efficient
algorithm for solving systems of nonlinear equations. Let the form of systems of nonlinear equations be
f1(x1,x2,...,xn)=0
f2(x1,x2,...,xn)=0
.
.
.
fn(x1,x2,...,xn)=0.
(1)
In order to transform (1) to an optimization problem, we will use the auxiliary function:
min f(x)=
n
i=1
f2
i(x), x=(x1,x2,...,xn). (2)
Corresponding author. Tel.: +98 914 116 2612; fax: +98 411 669 6012.
E-mail addresses: abdollahi_mm@yahoo.com,m.abdollahi89@ms.tabrizu.ac.ir (M. Abdollahi), isazadeh@tabrizu.ac.ir (A. Isazadeh),
abdollahi_d@daneshvaran.ac.ir (D. Abdollahi).
0898-1221/$ – see front matter ©2013 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.camwa.2013.04.018
M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908 1895
The equations system is reduced to the same form in the approach used in [3]. In Section 2, we describe the imperialist
competitive algorithm (ICA). In Section 3, some well-known systems are presented to demonstrate the effectiveness and
robustness of the proposed ICA. Then, in Section 4, we study some numerical tests. At the end, the conclusion is given in
Section 5.
2. Imperialist competitive algorithm
In this paper, we employ Imperialist competitive algorithm (ICA) to solve systems of nonlinear equations. Recently, a
number of methods have been proposed for solving systems of nonlinear equations such as genetic algorithms [4], particle
swarm algorithm [3]. ICA is a new evolutionary algorithm for optimizations which is inspired by imperialist competitive [5].
It is good to mention that ICA is a robust method based on imperialism which is the policy of extending the power and rule of
a government beyond its own borders [6]. In this algorithm, we start with an initial population as initial countries. Some of
the best countries among the population are selected to be the imperialists. The rest of the population is divided among the
mentioned imperialists as colonies. Then, the imperialistic competition begins among all the empires. The weakest empire
which cannot increase its power and is not able to succeed in this competition, will be eliminated from the competition. As a
result, all colonies move toward their relevant imperialists along with the competition among empires. Finally, the collapse
mechanism will hopefully cause all the countries to converge to a state in which there exists just one empire in the world
(in the domain of the problem), and all the other countries are colonies of that one empire. The robust empire would be our
solution.
2.1. Generating initial empires
Finding an optimal solution is the goal of optimization. We generate our countries which are the randomized solutions
as population [5]. In an N-dimensional problem, a country is an 1 ×Narray defined as follows:
country =(x1,x2,...,xn), xiR,1iN.(3)
We should generate Npop of them. The cost of each country is the cost of f(x)at the variables (x1,x2,...,xn). Then
cost =f(country)=f(x1,x2,...,xn). (4)
We select Nimp of the most powerful countries to form the empires. The remaining Ncol of the population will be the
colonies. As a result, we will have two types of countries: imperialist and colony. Now, we divide the Ncol colonies among
Nimp imperialists. We define the normalized cost of an imperialist by
Cn=cnmax
i{ci}(5)
where cnis the cost of nth imperialist and Cnis its normalized cost.
The normalized power of each imperialist is defined by
pn=
Cn
Nimp
i=1
Ci
(6)
So, the initial number of colonies of an empire will be
No.Cn=round(pn.Ncol)(7)
where No.Cnis the initial number of colonies of nth empire and Ncol is the number of all colonies. To divide the colonies for
imperialists, we randomly choose No.Cnof colonies to give them to the nth empire.
2.2. Moving the colonies of an empire toward the imperialist
Each colony that moves toward the imperialist by x-units in the direction is the vector from colony to imperialist. xwill
be a random variable with uniform distribution. Then
xU(0, β ×d), β > 1 (8)
where dis the distance between colony and imperialist. βcauses the colony to get closer to the imperialist. We have put
β=2 for all of our problems. (See Figs. 1 and 2.)
To get different points around the imperialist we have to add a random amount of deviation to the direction of movement
like θwhich is equal to 0.5 in this paper.
1896 M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908
Fig. 1. Moving colonies toward their relevant imperialist.
Source: From [5].
Fig. 2. Moving colonies toward their relevant imperialist in a randomly deviated direction.
Source: From [5].
2.3. Revolution
In each iteration, a number of colonies in an empire are replaced with the same number of new generated countries. We
have done this by generating some new countries and replacing them with some colonies of that empire, randomly. This
action is called revolution which has a sensitive role in this paper. The number of colonies of the empire which is supposed
to be replaced with the same number of new generated countries is:
N.R.C=round{RevolutionRate ×No.(The colonies of empiren)}(9)
where N.R.Cis the number of revolutionary colonies. This will improve the global convergence of the ICA and prevent it
sticking to a local minimum [7].
2.4. Exchanging positions of the imperialist and a colony
While moving a colony may access to a better position than that of the imperialist. So, the imperialist moves to the
position of that colony and vise versa.
2.5. Total power of an empire
The total power of an empire depends on its own all colonies as follows:
T·Cn=cost(imperialistn)+ξ·mean(cost(colonies of empiren)) (10)
where ξis a position coefficient. We have used the value of 0.02 in all of our problems.
2.6. Imperialistic competition
All empires are in competition with each other to take possession of colonies of other empires and control them. As a
result, the power of the weaker empires gradually begins to decrease and the power of more powerful ones increases. To
get to this goal, we find the possession probability of each empire based on its total power. The normalized total cost is
N·T·Cn=T·Cnmax{T·Ci}(11)
where T·Cnand N·T·Cnare respectively the total cost and the normalized total cost of nth empire. Now we could be able
to calculate the possession probability of each empire by
ppn=
N·T·Cn
Nimp
i=1
N·T·Ci
.(12)
M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908 1897
Table 1
Used parameters in ICA for tests and cases.
Parameter Value
Empires 10
RevolutionRate 0.02
ξ0.02
θ0.5
β2
Divide the mentioned colonies among empires, based on the possession probability of them. The vector Pis formed as
P= [pp1,pp2,pp3,...,ppNimp ](13)
and also the vector Rwith uniformly distributed elements
R= [r1,r2,r3,...,rNimp ]r1,r2,r3,...,rNimp U(0,1). (14)
Finally, we have vector Dby
D=PR= [pp1r1,pp2r2,pp3r3,...,ppNimp rNimp ].(15)
The elements of Dwill hand the mentioned colonies to an empire whose relevant index in Dis maximum.
2.7. The eliminated empire
When an empire loses all of its colonies, it will collapse and become one of the rest colonies.
2.8. Convergence
At the end, we will have the most powerful empire with no any competitor and all colonies will be under the control of
this unique empire. So, all the colonies will have the same costs as the unique empire has. It means that there is no difference
between colonies and their unique empire. In this ideal world, we put an end to our algorithm.
3. Proposed method
Since the proposed RevolutionRate in [7] is fixed during each process, so, in some problems, especially in the systems
of nonlinear equations, ICA falls in the local optimum. In this paper, to improve the efficiency of the algorithm, a similar
behavior to mutation in GA is simulated. Therefore, in each process, a random number on (0, 1) is produced. If it was less
or equal to RevolutionRate, the position of a colony randomly changes. Otherwise, does not change. In each iteration, this is
applied on each colony of an empire. This method raises the efficiency of ICA significantly.
4. Experiment and results
In this section, we have investigated the performance of ICA with four benchmark functions.
Test 1: 10 dimensions Rastrigin function
f(x)=
10
i=1
[x2
i10 cos(2πxi)+10] |xi| ≤ 5.2.
The solution is f(0,0,0,...,0)=0. Apply ICA to optimize it with 1000 iterations, and used parameters are shown in
Table 1 for the same 300 countries in [2].
The results of Mo et al. [2] and our final optimal results were given in Tables 2 and 3respectively. Fig. 3 shows the
convergence history of the ICA.
The results of ICA are better and we reached the optimized solution before 250 iterations with the same population size
used in [2].
Test 2: The Hartman’s function [2]
f(x)= −
4
i=1
ciexp
6
j=1
aij(xjpij )2,where 0 xj1,c=(1 1.233.2),
pij =
0.1312 0.1696 0.5569 0.0124 0.8283 0.5886
0.2329 0.4135 0.8307 0.3736 0.1004 0.9991
0.2348 0.1415 0.3522 0.2883 0.3047 0.6650
0.4047 0.8828 0.8732 0.5743 0.1091 0.0381
,pij =
10 3 17 3.5 1.7 8
0.05 10 17 0.1 8 14
3 3.5 1.7 10 17 8
17 8 0.05 10 0.1 14
1898 M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908
Table 2
Results of Mo et al.
Source: From [2].
Variables Initial iteration After 200 iterations After 400 iterations After 600 iterations After 800 iterations After 1000 iterations
x10.1431 0.0001 0.0007 0.0001 0.0000 0.0000
x22.1983 0.0001 0.0000 0.0001 0.0001 0.0001
x31.9401 0.0000 0.0000 0.0001 0.0000 0.0001
x41.7080 0.0002 0.0001 0.0000 0.0000 0.0000
x50.2261 0.9950 0.9962 0.9948 0.0001 0.0001
x60.9392 0.9950 0.9941 0.9949 0.9950 0.9949
x70.1129 0.9949 0.9949 0.0001 0.0001 0.0000
x80.1516 0.9950 0.9949 0.9949 0.9949 0.0000
x92.1893 0.0001 0.0000 0.0001 0.0000 0.0000
x10 4.9798 0.9950 0.0000 0.0001 0.0000 0.0000
Table 3
Results of Test 1 with ICA.
Variables Initial iteration After 200 iterations After 400 iterations After 600 iterations After 800 iterations After 1000 iterations
x12.883527 0.1084e007 0.1404e008 0.1404e008 0.1404e008 0.1404e008
x22.111072 0.1710e007 0.0275e008 0.0275e008 0.0275e008 0.0275e008
x30.869045 0.0048e007 0.0656e008 0.0656e008 0.0656e008 0.0656e008
x41.985114 0.6947e007 0.0855e008 0.0855e008 0.0855e008 0.0855e008
x51.156667 0.0328e007 0.1015e008 0.1015e008 0.1015e008 0.1015e008
x63.083374 0.0356e007 0.0899e008 0.0899e008 0.0899e008 0.0899e008
x73.093877 0.0948e007 0.0349e008 0.0349e008 0.0349e008 0.0349e008
x82.020172 0.1528e007 0.1610e008 0.1610e008 0.1610e008 0.1610e008
x92.832951 0.2304e007 0.0180e008 0.0180e008 0.0180e008 0.0180e008
x10 2.208695 0.2454e007 0.0147e008 0.0147e008 0.0147e008 0.0147e008
f(x)83.041615 1.3287e012 0000
Fig. 3. The convergence history of Rastrigin function (Test 1).
where min f(x)= −3.3220. The ICA was run 10 times and the parameters were same to Test 1 with 300 iterations. The
results of Mo et al. [2] and ours are shown in Tables 4 and 5respectively with the same parameters.
Test 3: Six-Hump camelback.
The Six-Hump camelback has six local optima, two of which are global.
min f(x)=4x2
12.1x4
1+1
3x6
1+x1x24x2
2+4x4
2.
The global solutions in [3] were
f(0.08984,0.71266)=f(0.08984,0.71266)= −1.0316285
with the convergence history shown in Fig. 4.
The ICA reached the best result with the same parameters in Table 1 quicker than PSO in [3] as follows
f(x1,x2)=f(0.089842012773979,0.712656402251958)= −1.031628453489878
and the convergence history is shown in Fig. 5 with the same 50 iterations.
M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908 1899
Table 4
Results of Mo et al.
Source: From [2].
Ordinal number Optimal solution (x1,x2,x3,x4,x5,x6)Optimal value Iteration iterations Mean iteration of 10 runs
1 (0.2031, 0.1479, 0.4767, 0.2753, 0.3116, 0.6573) 3.3220 79
137.5
2 (0.2030, 0.1469, 0.4758, 0.2756, 0.3120, 0.6572) 3.3220 74
3 (0.2019, 0.1455, 0.4766, 0.2754, 0.3112, 0.6573) 3.3220 227
4 (0.2022, 0.1475, 0.4772, 0.2752, 0.3115, 0.6568) 3.3220 86
5 (0.2018, 0.1468, 0.4774, 0.2755, 0.3122, 0.6582) 3.3220 221
6 (0.2031, 0.1479, 0.4767,0.2755, 0.3116, 0.6573) 3.3220 79
7 (0.2030, 0.1469, 0.4758, 0.2756, 0.3120, 0.6572) 3.3220 74
8 (0.2019, 0.5455, 0.4766, 0.2754, 0.3112, 0.6573) 3.3220 227
9 (0.2022, 0.1475, 0.4772, 0.2752, 0.3115, 0.6568) 3.3220 86
10 (0.2018, 0.1468, 0.4774, 0.2755, 0.3122, 0.6582) 3.3220 220
Table 5
Results of ICA.
Ordinal number Optimal solution (x1,x2,x3,x4,x5,x6)Optimal value Iteration iterations Mean iteration of 10 runs
1 (0.2023, 0.1458, 0.4753, 0.2754, 0.3118, 0.6574) 3.3220 47
83.3
2 (0.2021, 0.1475, 0.4756, 0.2760, 0.3115, 0.6574) 3.3220 88
3 (0.2012, 0.1467, 0.4785, 0.2755, 0.3119, 0.6570) 3.3220 89
4 (0.2017, 0.1467, 0.4784, 0.2752, 0.3117, 0.6573) 3.3220 97
5 (0.2014, 0.1455, 0.4771, 0.2750, 0.3112, 0.6573) 3.3220 72
6 (0.2017, 0.1472, 0.4763, 0.2746, 0.3116, 0.6572) 3.3220 96
7 (0.2030, 0.1469, 0.4774, 0.2758, 0.3115, 0.6573) 3.3220 85
8 (0.2004, 0.1470, 0.4759, 0.2748, 0.3119, 0.6575) 3.3220 73
9 (0.2026, 0.1471, 0.4754, 0.2750, 0.3117, 0.6572) 3.3220 87
10 (0.2016, 0.1468, 0.4785, 0.2756, 0.3112, 0.6574) 3.3220 99
Fig. 4. The convergence history of Six-Hump.
Source: From [3].
Test 4: This example was given in [3]
min f(x)=
D
i=1sin(xi)+sin 2xi
3.
The solution of this function is 1.21598D. The results of [3] and ICA for D=10 and D=100 are comparable in Tables 6–8
respectively. The variables in both algorithm were in (3, 13) and our number of countries are 300 as [3]. The ICA found the
optimal solution for D=10 before approximately 70 iterations and it found the optimal solution for D=100 before
approximately 700 iterations, much better than PPSO in [3].
See Figs. 6–8 too.
1900 M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908
Fig. 5. The convergence history of Six-Hump with ICA (Test 3).
Table 6
Results of Test 4.
Source: From [3].
Variables Initial iteration After 100 iterations After 200 iterations After 300 iterations After 400 iterations After 500 iterations
x14.9203 5.3737 5.3667 5.3656 5.3626 5.3623
x24.6815 5.3564 5.3601 5.3618 5.3628 5.3624
x34.9207 5.3522 5.3651 5.3658 5.3627 5.3621
x45.5048 5.3846 5.3656 5.3648 5.3636 5.3633
x56.3685 5.3597 5.3628 5.3630 5.3607 5.3627
x66.7112 5.3520 5.3669 5.3640 5.3631 5.3625
x75.6790 5.3369 5.3621 5.3626 5.3613 5.3624
x86.3557 5.3420 5.3626 5.3629 5.3623 5.3622
x911.7889 5.3705 5.3574 5.3647 5.3627 5.3627
x10 10.3531 5.3515 5.3580 5.3592 5.3622 5.3616
f(x)7.690599 12.158781 12.159769 12.159797 12.15981 12.15982
Table 7
The results of ICA for Test 4 with D=10.
Variables Initial iteration After 100 iterations After 200 iterations After 300 iterations After 400 iterations After 500 iterations
x17.648336 5.362271 5.362271 5.362271 5.362271 5.362271
x26.285073 5.362749 5.362749 5.362749 5.362749 5.362749
x36.441411 5.362276 5.362276 5.362276 5.362276 5.362276
x45.521101 5.362543 5.362543 5.362543 5.362543 5.362543
x56.255337 5.363662 5.363662 5.363662 5.363662 5.363662
x69.860261 5.362470 5.362470 5.362470 5.362470 5.362470
x79.630791 5.362061 5.362061 5.362061 5.362061 5.362061
x84.882258 5.362417 5.362417 5.362417 5.362417 5.362417
x95.152085 5.363256 5.363256 5.363256 5.363256 5.363256
x10 5.451308 5.361964 5.361964 5.361964 5.361964 5.361964
f(x)7.36443399 12.15982 12.15982 12.15982 12.15982 12.15982
Table 8
The results of Test 4 with D=100.
f(x)Initial
iteration
After 1000
iterations
After 2000
iterations
After 3000
iterations
After 4000
iterations
After 5000
iterations
After 6000
iterations
PPSO [3] 54.103342 121.208321 121.554754 121.593659 121.596941 121.598050 121.598204
ICA 29.786871 121.598200 121.598200 121.598200 121.598200 121.598200 121.598200
5. Case study
Six standard systems are selected from the literature to demonstrate the efficiency of the ICA for solving systems of
nonlinear equations.
Case 1: Geometry size of thin wall rectangle girder section
f1(x)=bh (b2t)(h2t)=165,b=The width of the section
M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908 1901
Fig. 6. The convergence history of Test 4 with D=10.
Source: From [3].
Fig. 7. The convergence history of ICA for Test 4 with D=10.
Fig. 8. The convergence history of ICA for Test 4 with D=100.
1902 M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908
Table 9
Results of Case 1.
Source: From [3].
Methods b h t f1(x)f2(x)f3(x)
PPSO (present study) 43.155566052654329 10.128950202278199 12.944048457756352 165 9369 6835
PPSO (present study) 7.602995198463455 24.541982377674739 11.576715672202731 165 9369 6835
Mo et al. [2] 8.943089 23.271482 12.912774 251.2378 9369 6835
Luo et al. [1] 12.5655 22.8949 2.7898 408.6488 9369 6835
Luo et al. [1]12.5655 22.8949 2.7898 408.6488 9369 6835
Luo et al. [1] 8.943089 23.271482 12.912774 251.2378 9369 6835
Luo et al. [1]8.943089 23.271482 12.912774 251.2378 9369 6835
Luo et al. [1]2.3637 35.7564 3.0151 334.0376 9369 6835
Luo et al. [1] 2.3637 35.7564 3.0151 334.0376 9369 6835
Table 10
Comparison results of ICA for Case 1 with [1–3].
Methods b h t f1(x)f2(x)f3(x)
ICA (present study) 8.943088778747601 23.271481879207862 12.912774291361677 165 9369 6835
PPSO [3] 43.155566052654329 10.128950202278199 12.944048457756352 709.2412 9369 6835
PPSO [3]7.602995198463455 24.541982377674739 11.576715672202731 208.1851 9369 6835
Mo et al. [2] 8.943089 23.271482 12.912774 165 9369 6835
Luo et al. [1] 12.5655 22.8949 2.7898 166.7229 9369 6835
Luo et al. [1]12.5655 22.8949 2.7898 166.7229 9369 6835
Luo et al. [1] 8.943089 23.271482 12.912774 165 9369 6835
Luo et al. [1]8.943089 23.271482 12.912774 165 9369 6835
Luo et al. [1]2.3637 35.7564 3.0151 165 9369 6835
Luo et al. [1] 2.3637 35.7564 3.0151 165 9369 6835
f2(x)=bh3
12 (b2t)(h2t)3
12 =9369,h=The height of the section
f3(x)=2t(ht)2(bt)2
h+b2t=6835,t=The thickness of the section.
The results in [3] were printed incorrectly as shown in Table 9. The best solutions obtained by the ICA method have been
listed in Table 10 and compares them with correct results reported by Mo et al. [2] and Luo et al. [1]. It is obvious from
Table 10 that the results of the ICA method outperform other three results with the same 300 iterations and 250 countries
as population and other parameters are shown in Table 1.
Case 2:
xx2
1+xx1
25x1x2x3=85
x3
1xx3
2xx2
3=60
xx3
1+xx1
3x2=2
3x15,2x24,0.5x32.
The solution in [2,3] was (4, 3, 1). The ICA method got the same result but the convergence history of ICA is better with
300 iterations and 250 countries while [3] had been reached with 1000 iterations and 250 population to the answer. See
Figs. 9 and 10.
Case 3:
x3
13x1x2
21=0
3x2
1x2x3
2+1=0.
The solutions in [3,8] were
f(0.29051455550725,1.08421508149135)=4.686326815078573e 029
f(0.793700525984100,0.793700525984100)=1.577721810442024e 030
with 120 iterations and unknown number of population. The results of the ICA method are
f(1.084215081491351,0.290514555507251)=3.562200025138631e 030
f(0.793700525984100,0.793700525984100)=1.577721810442024e 030
f(0.290514555507251,1.084215081491351)=3.562200025138631e 030
with 50 iteration and 250 countries. Figs. 11 and 12 shows the convergence history of Case 3.
M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908 1903
Fig. 9. The convergence history of Case 2.
Source: From [3].
Fig. 10. The convergence history of Case 2 with ICA.
Case 4: Neurophysiology Application
x2
1+x2
3=1
x2
2+x2
4=1
x5x3
3+x6x3
4=0
x5x3
1+x6x3
2=0
x5x1x2
3+x6x2
4x2=0
x5x2
1x3+x6x2
2x4=0
10 xi10,1i6.
We considered the example proposed in [9,10]. The best known solution in [9] among 12 different solutions has been
shown in Table 11 beside the exact solution of ICA with the same 300 countries and 200 iterations in [9].
The convergence history of Case 4 is shown in Fig. 13.
Case 5: (Problem 2 in [11] and Test Problem 14.1.4 in [12])
0.5 sin(x1x2)0.25x20.5x1=0
(10.25)(exp(2x1)e)+ex22ex1=0
0.25 x11,1.5x22π.
1904 M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908
Fig. 11. The convergence history of Case 3.
Source: From [3].
Fig. 12. The convergence history of Case 3 with ICA.
Table 11
Comparison results of Case 4.
The best results of [9] The results of ICA
Variables values Functions values Variables values Functions values
0.8078668904 0.0050092197 0.041096050919063 0
0.9560562726 0.0366973076 0.041096050919063 0
0.5850998782 0.0124852708 0.999155200456294 0
0.2219439027 0.0276342907 0.999155200456294 0
0.0620152964 0.0168784849 0.098733550533454 0
0.0057942792 0.0248569233 0.098733550533454 0
The known solution of Case 5 in [11] is
x=(0.50043285,3.14186317)
f=(0.00023852,0.00014159)=7.693745216994211e 008.
The known solutions of Case 5 in [12] are (0.29945, 2.83693) and (0.5, 3.14159).
The results of the ICA method with 250 iterations and 250 countries as [11] are
x=(0.299448692495720,2.836927770471037)
f=(1.305289210051797e 012,2.284838984678572e 013)=5.631272867601562e 024
M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908 1905
Fig. 13. The convergence history of Case 4 with ICA.
Fig. 14. The convergence history of Case 5 with ICA.
and
x=(0.500000000000000,3.141592653589794)=1
2, π
f=(0,0)=0.
Fig. 14 shows the convergence history of Case 5.
Case 6: (Problem 6 in [11] and Test Problem 14.1.6 in [12]).
This problem has been solved by the filled function method in [11] and proposed problem in [12].
4.731 ×103x1x30.3578x2x30.1238x1+x71.637 ×103x20.9338x40.3571 =0
0.2238x1x3+0.7623x2x3+0.2638x1x70.07745x20.6734x40.6022 =0
x6x8+0.3578x1+4.731 ×103x2=0
0.7623x1+0.2238x2+0.3461 =0
x2
1+x2
21=0
x2
3+x2
41=0
x2
5+x2
61=0
x2
7+x2
81=0
1xi1,i=1,...,8.
The known solution of Case 6 in [11,12] and our results are shown in Table 12 with 1000 iterations and 300 countries.
(See Fig. 15.)
1906 M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908
Table 12
Comparison results of Case 6.
Method xVariables values fFunctions values
The best in [11]
x10.67154465 f10.00000375
x20.74097111 f20.00001537
x30.95189459 f30.00000899
x40.30643725 f40.00001084
x50.96381470 f50.00001039
x60.26657405 f60.00000709
x70.40463693 f70.00000049
x80.91447470 f80.00000498
The best in [12]
x10.1644 f18.8531e005
x20.9864 f23.5894e005
x30.9471 f36.6216e006
x40.3210 f42.1560e005
x50.9982 f51.2320e005
x60.0594 f63.9410e005
x70.4110 f76.8400e005
x80.9116 f86.4440e005
The best of ICA
x10.164431665854327 f12.775557561562891e016
x20.986388476850967 f21.110223024625157e016
x30.718452601027603 f31.734723475976807e018
x40.695575919707312 f41.665334536937735e016
x50.997964383970433 f50
x60.063773727557003 f60
x70.527809105283546 f70
x80.849363025083964 f80
Fig. 15. The convergence history of Case 6 with ICA.
5.1. Discussion
There is a diverse range of mathematical methods and evolutionary algorithms for optimization problems especially
for solving systems of nonlinear equations. In this paper, the efficiency of ICA for optimization of different examples
are compared to different methods such as Hybrid Approach with Chaos Optimization and Quasi-Newton [1], Conjugate
Direction Particle Swarm Optimization (CDPSO) [2], Proposed Particle Swarm Optimization (PPSO) [3], Genetic Algorithm
(GA) [9], A New Filled Function Method [11] and Homotopies Exploiting Newton Polytopes [10]. In all results, ICA
outperforms other mentioned methods with less iteration than the other discussed methods. For example, we reached the
exact solution of Test 1 with 400 iterations in comparison to [2] with 1000 iterations. Table 8 shows results for a large scale
problem which ICA performs well.
The efficiency of the proposed method is due to manipulation of the revolution policy of ICA. We implement the similar
strategy to mutation as a revolution in ICA [13]. This significantly improves the performance of ICA. The statistical results of
tests and cases with 30 independent runs in Table 13 show the stability and convergence of our proposed method. We use
One-Sample t-test for a comparison of the average of cases (observed averages) and the countries (expected averages) with
an adjustment for our five cases in the sample and the standard deviation of the average (See Table 14).
M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908 1907
Table 13
Statistical results.
Problem NMean Std. deviation Std. error mean Worst Best
Test 1 30 0.0 0.0 0.0 0.0 0.0
Test 2 30 3.305370000000002E0 5.172567660929779e2 9.443773293709911e33.1299 3.3220
Test 3 30 1.031628453489877E0 4.903400625422253e16 8.952343770095692e17 1.031628453489877 1.031628453489878
Test 4 (D=10) 30 1.2159820457168102E1 4.351616938118602e07 7.944929195430403e08 1.2159821619352311E1 1.2159820006065519E1
Test 4 (D=100) 30 1.215982007433365E2 1.131969231964622e6 2.066683609162738e7 1.215982058451825E2 1.215982000134926E2
Case 1 30 3.301176194526734e14 1.808127293357038e13 3.301173684708036e14 9.903521305104092e013 2.252128464646329e024
Case 2 30 8.948499236529537e18 4.580698621477245e17 8.363173213719686e18 2.513443969185863e016 0.0
Case 3 30 0.0 0.0 0.0 0.0 0.0
Case 4 30 2.970867386475955e18 1.615503654126465e17 2.949492643656963e18 8.850441823038988e017 0.0
Case 5 30 1.145605502924358e15 6.269216037417460e15 1.144597013855565e15 3.433890687251408e014 0.0
Case 6 30 5.560518602264908e25 2.527736286739731e24 4.614993945572325e25 1.378113375386532e023 1.170995498842820e031
Table 14
One-sample t-test results.
95% Confidence interval of the difference
Problems tdf Sig. (2-tailed) Mean difference Lower Upper H0in level α=0.05
Case 1 1.000 29 0.326 3.301176194526734e14 3.450482079265911e14 1.005283446831938e13 Accepted
Case 2 1.070 29 0.293 8.948499236529537e18 8.156110522458490e18 2.605310899551756e17 Accepted
Case 4 1.007 29 0.322 2.970867386475955e18 3.061522397583018e18 9.003257170534929e18 Accepted
Case 5 1.001 29 0.325 1.145605502924358e15 1.195358238109387e15 3.486569243958104e15 Accepted
Case 6 1.205 29 0.238 5.560518602264908e25 3.878203813481636e25 1.499924101801145e24 Accepted
1908 M. Abdollahi et al. / Computers and Mathematics with Applications 65 (2013) 1894–1908
6. Conclusions and future works
This paper proposes a new efficient approach for solving systems of nonlinear equations. The system of nonlinear
equations was transformed into a multi-objective optimization problem. The goal was to obtain values as close to zero
as possible for each of the involved objectives. Some well-known problems were presented to demonstrate the efficiency of
the Imperialist Competitive Algorithm (ICA) in comparison with other algorithms such as PPSO, CDPSO, GA, Filled Function
Method and Homotopies Exploiting Newton Polytopes. This paper aims to improve the revolution policy of ICA as mentioned
in Section 3. Therefore, the proposed method reached more accurate solutions than the other methods. As a future work,
we are planning to extend ICA on solving the boundary value problems such as Harmonic and Biharmonic equations.
Furthermore, the normal distribution can be used instead of uniform distribution to achieve better results. It is noteworthy
that the convergence speed could be raised by the use of chaos theory for θ[14].
Acknowledgments
The authors would like to acknowledge Mr. E. Atashpaz Gargari for the package of ICA used in this work, and specially
thank Miss S. Seifollahi for helping to put statistical data.
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The integrated product mix-outsourcing optimization is a major problem in manufacturing enterprise. Generally, heuristic or meta-heuristic solution approaches are used to optimize such problems. Heuristic approaches for these problems include Theory of Constraints (TOC) and Standard Accounting. Sometimes heuristic approaches are inefficient especially in large problems and instead, in these cases meta-heuristic algorithms have been applied extensively. In this paper a novel meta-heuristic algorithm “Imperialist Competitive Algorithm” (ICA) is applied to solve the integrated product mix-outsourcing optimization problem. Also, the results obtained from ICA are compared with the results of TOC and Standard Accounting approaches.
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Solving systems of nonlinear equations is one of the most difficult numerical computation problems. The convergences of the classical solvers such as Newton-type methods are highly sensitive to the initial guess of the solution. However, it is very difficult to select good initial solutions for most systems of nonlinear equations. By including the global search capabilities of chaos optimization and the high local convergence rate of quasi-Newton method, a hybrid approach for solving systems of nonlinear equations is proposed. Three systems of nonlinear equations including the “Combustion of Propane” problem are used to test our proposed approach. The results show that the hybrid approach has a high success rate and a quick convergence rate. Besides, the hybrid approach guarantees the location of solution with physical meaning, whereas the quasi-Newton method alone cannot achieve this.