1. International Journal of Electronics and JOURNALEngineering & Technology (IJECET), ISSN 0976 –
INTERNATIONAL Communication OF ELECTRONICS AND
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 5, September – October (2013), © IAEME
COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
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ISSN 0976 – 6472(Online)
Volume 4, Issue 5, September – October, 2013, pp. 218-224
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SELF ACCELERATED SMART PARTICLE SWARM OPTIMIZATION FOR
NON LINEAR PROGRAMMING PROBLEMS
Anuradha L. Borkar1
1
1
Electronics and Telecommunication Dept.
M S S S’ College of Engineering and Technology, Jalna, Maharashtra, India
ABSTRACT
This paper presents Self Accelerated Smart Particle Swarm Optimization (SASPSO) for
nonlinear programming problems (NLP). In SASPSO, the positions of particle are updated by pbest
(Personal or local best) and gbest (Global best). The main advantages of SASPSO are that it doesn’t
require velocity equation. In addition, it does not require any additional parameter like acceleration
coefficients and inertia weight as in case other PSO algorithms. Momentum factor is introduced in
SASPSO which can prevent particles from out of defined region without checking the validity of
positions at every iterations result in saving of computational cost. During the initial stages of the
experimentation, the step size will be large and during the final stage of the experimentation, the step
size is reduced to smaller value. The SASPSO is tested on global optimization problems such as
Nonlinear Programming Problems (NLP). The results are compared with the Genetic Algorithm
(GA). The results of SASPSO are good in terms of are accuracy of optimal solution and generations
as compared GA. It also gives large number of possible optimal solutions as compared GA.
Keywords: gbest(Global best), Nonlinear programming problems (NLP), pbest (Personal or local
best), Self Accelerated Smart Particle Swarm Optimization (SASPSO).
I.
INTRODUCTION
Most of the real life problems occurring in the field of science and engineering may be
modeled as Nonlinear programming (NLP), unimodal or multimodal optimization problems.
Nonlinear programming is the process of solving a system of equalities and inequalities over a set of
unknown real variables, along with an objective function to be maximized or minimized, where some
of the constraints or the objective functions are nonlinear. No general algorithms exist for solving
nonlinear programming problems. However for problems with certain suitable structures, efficient
algorithms have been developed. It is possible to convert the given NLP into one in which these
structures become visible.
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2. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 5, September – October (2013), © IAEME
Nonlinear programming problems, unimodal or multimodal problems are generally
considered more difficult to solve as there exists several local and global optima. Effective
optimization of nonlinear programming problems, multi-dimensional and multimodal function with
faster convergence and good quality of solution is a challenging task. The high computational cost
and demands for improving accuracy of global optima of nonlinear programming problems and
multimodal functions have forced the researchers to develop efficient optimization techniques in
terms of new, modified or hybrid soft computing techniques.
John Holland conceived Genetic Algorithm (GA) in the mid 1970’s [1]. GA is inspired by the
principles of genetic evolution and mimics the reproduction behavior observed in biological
populations. GA suffers from premature convergence when an individual that is fit than others at
early stages dominates on the reproduction process leading to a local optimum convergence rather
than a more thorough search that could have lead to a global optimum [1]. GA has been extensively
applied to solve complex design optimization problems due its capability to handle constrain
functions without requiring gradient information [1]. But the limitation of getting trapped in local
minima and three step procedures of GA [1-2] such as selection, crossover and mutation increase the
computational time and thus forced the researchers to search for more efficient optimization
techniques.
Particle Swarm Optimization (PSO) is a population-based optimization method developed by
Eberhart and Kennedy in 1995. This method is inspired by social behavior of bird flocking or fish
schooling. It can efficiently handle problems like nonlinear, unimodal and multimodal function
optimizations [3]. . Compared to GA, PSO is easy to implement and converge faster with less
memory requirement [4]. However, unlike GA, PSO has no evolution operators such as crossover
and mutation.
The new variants of PSO are proposed for faster convergence and better quality of optimum
solution like Supervisor-Student Model in Particle Swarm Optimization (SSM-PSO) [5], Linear
Decreasing Weight Particle Swarm Optimization (LDW-PSO) [6], Gregarious Particle Swarm
Optimization (GPSO) [7], Global and Local Best Particle Swarm Optimization (GLBestPSO)[8] and
Emotional Particle Swarm Optimization (EPSO)[9].
The author proposed Self Accelerated Smart Particle Swarm Optimization (SASPSO) for
nonlinear programming problems (NLP). The rest of the paper is organized in four fold. The section
II depicts the review of original Particle Swarm Optimization (PSO). The section III depicts the
proposed method. In section IV, experimental results on nonlinear programming problems (NLP).by
proposed method and other published techniques are presented. Section V comprises of conclusion.
II
REVIEW OF PARTICLE SWARM OPTIMIZATION
The original framework of PSO is designed by Kennedy and Eberhart in 1995. There fore, it
is known as standard PSO [3].
PSO follows the optimization process by means of personal or local best (pi), global best
(pg), particle position or displacement (X) and particle velocity (V). For each particle, at the current
time step, a record is kept for the position, velocity, and the best position found in the search space.
Each particle memorizes its previous velocity and the previous best position and uses them in its
movements [3]. The velocities (V) of the particles are limited in [Vmin Vmax] D. If V is smaller
than Vmin then V is set to Vmin or Xmin. If V greater than Vmax then V is set to Vmax or Xmax.
Since the original version of PSO lacks velocity control mechanism, it has a poor ability to search at
a fine grain.
The two updating fundamental equations in a PSO are velocity and position equations, which
are expressed as Eq. (1) and (2).
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3. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 5, September – October (2013), © IAEME
V id (t + 1) = V id ( t ) + c1 ∗ r1 d ( t ) ∗ ( p id ( t ) − X id ( t )) + c 2 ∗ r2 d (t ) ∗ ( p gd (t ) − X id ( t ))
(1)
X id (t + 1) = X id (t ) +V id (t + 1)
(2)
Where,
t= Current iteration or generation.
i = Particle Number.
d= Dimensions.
Vid(t) = Velocity of i-th particle for d-dimension at iteration t.
Xid (t) = Position of i-th particle for d-dimension at iteration t.
c1 and c2= Acceleration constants.
r1d (t) and r2d (t) = Random values [0 1] for d- dimension at iteration t.
pid (t)= Personal or local best of i-th particle for d-dimension at iteration t.
pgd (t) = Global best for d-dimension at iteration t.
The right side of Eq. (1) consists of three parts. The first part of equation is the previous
velocity of the particle. The second part is the cognition (self-knowledge) or memory, which
represents that the particle is attracted by its own previous best position and moving toward to it. The
third part is the social (social knowledge) or cooperation, which represents that the particle is
attracted by the best position so far in population and moving towards to it. There are restrictions
among these three parts and can be used to determine the major performance of the algorithm.
III
SELF ACCELERATED SMART PARTICLE SWARM OPTIMIZATION (SASPSO)
The standard PSO suffers from following disadvantages.
1.
2.
3.
4.
The swarm may prematurely converge when some poor particles attract the other particles or
due to local optima or bad initialization
It has problem dependent performance. No single parameter settings exists which can be
applied to all problems.
It requires tuning of parameters like c1, c2, w, iterations and swarm size
Increasing the value of inertia weight w, increases the speed of the particles resulting in more
global search and less local search. Decreasing the inertia weight slows down the speed of
the particle resulting in more local search and less global search.
So to avoid the disadvantage of standard PSO, the author proposed a Self Accelerated Smart
Particle Swarm Optimization (SASPSO). In SASPSO, the positions of particle are updated by pbest
(Personal or local best) and gbest (Global best particle positions) as expressed in Eq. (3). The main
advantages of SASPSO are that it doesn’t require velocity equation. In addition, it does not require
any additional parameter like acceleration coefficients and inertia weight as the case in other PSO
algorithms. Momentum factor can prevent particles from out of defined region without checking the
validity of positions at every iterations result in saving of computational cost.
Xi(t+1)= Xi(t)+Mc*rand((Xi(t)- (gbest- pbesit))+( gbest-- pbesit)*rand)
Where,
t= Current iteration or generation.
i = Particle Number.
Xi (t) = Position of i-th particle for d-dimension at iteration t.
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(3)

4. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 5, September – October (2013), © IAEME
r1 and r2 = Random values [0 1] at iteration t.
pbesit = Personal or local best of i-th particle at iteration t.
Gbest = Global best at iteration t.
Mc= Momentum Factor
During the initial stages of the experimentation, the step size will be large and thus the
positions of particles are away from the global best position. During the final stage of the
experimentation, the step size is reduced to smaller value. Momentum factor can prevent particles
from out of defined region without checking the validity of positions at every iterations result in
saving of computational cost.
IV
EXPERIMENTAL RESULTS AND DISCUSSIONS
The Four Quadratic Programming Problems [10] as shown in Table 1 are used to validate
performance of the Self Accelerated Smart Particle Swarm Optimization. The authors are considered
both maximization and minimization functions to test efficiency of the SASPSO. The first and
second problems are minimization problem and third and fourth problem is maximization problem.
The solution by using Beale’s method [10] for each Quadratic Programming problem is presented in
Table 1. The equation along with the constraints is given to the program. The objective function
which is to be minimized or maximized is the fitness function.
In SASPSO, population is taken as double vector with size 20. The value of momentum
factor is chosen as 0.005 that will results in not escaping the global optima. The stopping criterion of
the SASPSO is taken as error value of 0.1. If the error value is equal to or less than 0.1 then the
SASPSO is stopped. The number of trials is taken as 10.
The number of generations required and solutions for Quadratic Programming problem by
using GA and SASPSO is presented in Table 2. For first and second NLP problems SASPSO is
required less number of generations as compared GA and the accuracy of solutions by SASPSO
which is better than GA. For third and fourth NLP problems SASPSO is required less number of
generations as compared GA and the accuracy of solutions by SASPSO is 100 and which is better
than GA. But the SASPSO is producing same solution for maximization problem. The Fig 1 shows
number of generations required for each NLP problems by GA and SASPSO. As seen from results
the SASPSO requires less number of computations as compared to GA improves the quality of
optima . The presented method is suitable for optimization of Quadratic Programming problem,
unimodal and multimodal functions.
Eq No
1
2
3
4
Table 1. Quadratic Programming Problems solved
Equation With Constraints
Beale’s method
Min Z= X1*2+ X2^2
X1=2
SUB: X1+ X2>=4
X2=2
2 X1+ X2>=5 X1, X2 >=0
Z=8
Min Z=183-44 X1-42 X2+8 X1^2-12 X1 X2^2
X1=3.8
SUB: 2 X1+ X2<=10
X2=2.4
X1, X2>=10
Z=19
Max Z= 2 X1+3 X2
X1=2
SUB: X1^2+ X2^2<=20
X2=4
X1 X2<=8
Z=16
X1, X2>=0
Max 2 X1+2 X2-2 X2^2
X1=2
SUB : X1+4X2<=4
X2=0
X1+ X2<=2
Z=4
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5. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 5, September – October (2013), © IAEME
Table 2. Comparison of Quadratic Programming Problems solved by using GA and SASPSO
Eq
No
Solution by GA[10]
Gen.
[10]
Optimize
using
GA[10]
Accu
racy
[10]
Gen.
Optimize
using
SASPSO
Accuracy
3
8.00208
100
X1
1.90715
4
8.01732
100
2.15473
2.05038
1.84527
1.94965
4
4
8.04788
8.00519
100
100
1.95916
1.95270
2.05151
2.04818
3
3
8.00470
8.00081
100
100
2.14182
2.01181
1.8582
1.98821
4
4
8.0403
8.00038
100
100
1.95688
1.96214
2.05713
2.05779
3
3
8.06115
8.08449
100
100
1.95293
1.95116
2.05772
2.05428
3
3
8.00048
8.02708
100
100
1.94477
1.95201
1
X2
2.09287
Solution by
SASPSO
X1
X2
1.95539
2.04873
2.05895
2.06040
3
3
8.02144
8.00555
100
100
2.05079
2.36760
3
3
8.00377
19.00165
100
100
3.81998
5
19.002
100
3.80427
3.79966
2.39102
2.40058
6
4
19.00247
19.00061
100
100
3.80799
3.81026
2.37445
2.37094
3
3
19.00714
19.00020
100
100
3.79831
3.80224
2.40337
2.39492
8
7
19.00035
19.0042
100
100
3.79655
3.81848
2.39886
2.35490
4
6
19.00018
19.00029
100
100
3.82026
3.80526
2.35451
2.3736
7
6
19.00793
19.00008
100
100
3.79655
3.80526
2
2.36005
1.95908
3.81169
2.39886
2.37368
7
5
19.00048
19.00108
100
100
2.35260
4.00526
7
2
19.09185
16.02632
100
100
2.09653
5
15.64055
97.75
1.99
1.91904
4.00379
4.039
3
4
15.99137
15.95508
99.94
99.71
2.00526
2.00526
4.00526
4.00526
2
2
16.02632
16.02632
100
100
1.72685
1.9912
4.12527
4.00436
5
5
15.82953
15.99548
98.93
99.97
2.00526
2.00526
4.00526
4.00526
2
2
16.02632
16.02632
100
100
2.00526
2.00526
4.00526
4.00526
2
2
16.02632
16.02632
100
100
2.00526
2.00526
3
3.81583
3.82048
2.00526
4.00526
4.00526
2
2
16.02632
16.02632
100
100
1.95734
3
3.99635
99.9
2.00526
2.00500
4.00526
0.00100
2
2
16.02632
4.01300
100
100
1.97025
1.86849
4
0.04266
0.02975
0.13161
3
3
3.99823
3.9975
99.95
99.93
2.00500
2.00500
0.00100
0.00100
2
2
4.01300
4.01300
100
100
1.94342
1.89092
0.05658
0.10902
3
3
3.9936
3.9761
99.93
99.4
2.00500
2.00500
0.00100
0.00100
2
2
4.01300
4.01300
100
100
2.00500
2.00500
0.00100
0.00100
2
2
4.01300
4.01300
100
100
2.00500
2.00500
0.00100
0.00100
2
2
4.01300
4.01300
100
100
2.00500
0.00100
2
4.01300
100
222

6. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 5, September – October (2013), © IAEME
Number of Generations
Comparision of Number of Generations required by PSO
and SASPSO
7
6
5
4
3
2
1
0
PSO
SASPSO
1
2
3
4
Equation Number
Fig. 1. Number of Generations Required by PSO and SASPSO
V.
CONCLUSION
The author introduced Self Accelerated Smart Particle Swarm Optimization (SASPSO) to
optimize difficult nonlinear programming problems (NLP) with fast convergence and better
accuracy.. The main advantages of SASPSO are that it doesn’t require velocity equation. In addition,
it does not require any additional parameter like acceleration coefficients and inertia weight as in
case other PSO algorithms. Momentum factor is introduced in SASPSO which can prevent particles
from out of defined region without checking the validity of positions at every iterations result in
saving of computational cost. Using SASPSO, we get large combinations of values of decision
variables satisfying all the given constraints in very short time. The results of SASPSO are good in
terms of are accuracy of optimal solution with less number of generations as compared GA. The
proposed technique is more efficient for improving the quality of global optima of Nonlinear
programming problems (NLP), unimodal and multimodal functions with less computational
requirement and better accuracy.
VI.
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