# A Local and Global Search Combined Particle Swarm Optimization

Authors: Weitian Lin; Zhigang Lian; Xingsheng Gu; Bin Jiao
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

Particle swarm optimization algorithm (PSOA) is an advantage optimization tool.
However, it has a tendency to get stuck in a near optimal solution especially
for middle and large size problems and it is difficult to improve solution
accuracy by fine-tuning parameters. According to the insufficiency, this paper
researches the local and global search combine particle swarm algorithm
(LGSCPSOA), and its convergence and obtains its convergence qualification. At
the same time, it is tested with a set of 8 benchmark continuous functions and
compared their optimization results with original particle swarm algorithm
(OPSOA). Experimental results indicate that the LGSCPSOA improves the search
performance especially on the middle and large size benchmark functions
significantly.


---

## Body

## 1. Introduction

Swarm intelligence (SI) techniques, including particle swarm optimization (PSO) and
ant colony optimization (ACO), use swarm behavior to solve the problem, and they use
the concept of neighborhood intelligence along with individual intelligence.
Particle swarm was originally formulated to study the interesting concept of a
social behavior (bird flocking or fish schooling) in a simplified manner in
simulation. However, very soon the potential of this technique was realized to
develop into a powerful optimization tool which can be successfully applied in the
fields of both continuous and discrete optimization problems such as function
optimization, scheduling and hence received many attentions since it is
originated.The particle swarm concept originated as a simulation of simplified social system,
and this technique has been developing rapidly. Eberhart and Kennedy in [1, 2]
originally formulated particle swarm in 1995, and recently, several investigations
have been undertaken to improve the performance of original PSO. There are many
improved PSO algorithm such as Shi and Eberhart who introduced an inertia weight
making the influence of the previous velocity on the new velocity in [3–5] and Kennedy and Mendes use cluster analysis improving PSO’s
performance in [6]. Angeline presented the
model of improved PSO algorithm called (HPSO) in [7] which uses selection operation of evolution computing. Literature
[8] Suganthan sets up the model of PSO
with adjacent filed operation. Jiao et al. present a dynamic inertia weight particle
swarm optimization algorithm in paper [9] in
which the dynamic inertia weight that decreases according to iterative generation
increasing is used. Many experts also studied others properties of PSO such as Clerc
and Kennedy in [10, 11] who have researched the particle swarm explosion stability
and convergence in a multidimensional complex space. Eberhart and Shi have
investigated PSO developments, applications, and resources in [12] and parameter selection in [13] and dimension selection methods in [13].PSO has been successfully applied in many areas: function optimization, artificial
neural network training, fuzzy system control, and other areas where GA can be
applied. Parsopoulos and Vrahatis approach global optimization problems using PSO in
[14, 15] and Abido uses PSO optimizing power flow in [16]. PSO technique have been used in assignment problem in
[17], reactive power and voltage control
considering voltage security assessment in [18], and so forth. PSO algorithm is widely used, such as the paper
[19, 20] that introduces a Hybrid Particle Swarm algorithm with artificial
Immune Learning (HPSIL) for solving Fixed Charge Transportation Problem (FCTP).
Hamta et al. in paper [21] presented a hybrid
PSO algorithm for a multiobjective assembly line balancing problem with flexible
operation times, sequence-dependent setup times, and learning effect. Chou [22] used a particle swarm optimization with
cocktail decoding method for hybrid flow shop scheduling problems with
multiprocessor tasks. This study searches for a number of solid decoding methods
that can be incorporated into the cocktail decoding method. Then, it develops a
particle swarm optimization (PSO) algorithm that can be combined with the cocktail
decoding method. PSO has particularly gained prominence due to its relative ease of
operation and capability to quickly arrive at an optimal/near-optimal solution.
However it was pointed out in our studies that original PSO might frequently tend to
get stuck in a near optimal solution in reaching optimum solutions especially for
middle or large size function optimization problems. According to the shortcoming of
original PSO, a local and global search combine particle swarm optimization
algorithm is presented in this paper, and the simulation results show that it is
efficacious. At the same time, this paper rigorously analyzes new PSO algorithm
convergence and obtains its convergence qualification.This work differs from the existing ones at least in four aspects: firstly, it
proposes a local and global search combine particle swarm optimization (LGSCPSO)
algorithm iterative formula, in which all particles share the best information of
the local particles, global particles and neighborhood particles. Secondly, it finds
the best combine parameters of LGSCPSO for different size optimization problems.
Thirdly, it is to compare the original PSO with LGSCPSO and shows that the latter is
more efficacious for optimization problems. Fourthly, it strictly analyzes the PSO
algorithm’s convergence and obtains its convergence qualification. The rest
of the paper is organized as follows: The next section introduces the original PSO
model. The iteration formulation of LGSCPSO algorithm is presented in Section3. In Section 4, it analyzes the PSO algorithm’s convergence and finds its
convergence qualification. Then, it describes the test functions, experimental
settings, and compares experimental results of original PSO with LGSCPSO algorithm.
Finally, Section 5 summarizes the
contribution of this paper and conclusions.


## 2. Original Particle Swarm Optimization Algorithm

### 2.1. Description of Original Particle Swarm Optimization Algorithm

Particle swarm optimization (PSO) is an evolutionary computation technique, and
its concept originated as a simulation of a simplified social system. The major
difference is that the evolution computing techniques use genetic operators
whereas swarm intelligence techniques use the physical movements of the
individuals in the swarm. PSO is distinctly different from other
evolutionary-type methods in a way that it does not need complex encoding and
decoding processes and does not use the operation (such as crossover and/or
mutation), and it takes real numbers as particles in the aspects of
representation solution. From the procedure, one can learn that PSO shares many
common points with GA; for example, they start with a neighborhood of a randomly
generated population, evaluate the population with fitness values, update the
population and search for the optimum with random techniques, and do not
guarantee optimal. Similar to other population-based algorithms, PSO as an
optimization tool can solve a variety of difficult optimization problems.
Compared to GA, one of the important attractive factors of PSO is simple that
there are very few parameters to adjust. It can achieve the optimal or
near-optimal solutions in a rather short time without enormous iterative
computations in digital implementation.Swarm intelligence simulates a social behavior such as bird flocking to a
promising position for certain objectives in a multidimensional space. It is
initialized with a population of random solutions and searches for optima by
updating generations. PSO system combines local search method (through
self-experience) with global search methods (through neighboring experience),
attempting to balance exploration. Like evolutionary algorithm, PSO search uses
a population (called swarm) of individuals (called particles) that are updated
from iteration to iteration. Each particle represents a candidate position
(i.e., solution) to the problem at hand, resembling the chromosome of GA. A
particle is treated as a point in anM-dimension space, and the
status of a particle is characterized by its position and velocity. Initialized
with a swarm of random particles, PSO is achieved through particle flying along
the trajectory that will be adjusted based on the best experience or position of
the one particle (called local best) and ever found by all particles (called
global best). PSO updates a population of particles with the internal velocity
and attempts to profit from the discoveries of themselves and previous
experiences of all other companions.


### 2.2. The Model of Original Particle Swarm Optimization Algorithm

In every search iteration, each particle is updated by following two
“best” values. The first one is the best solution (fitness) it has
achieved so far, and this value is calledp
Best. Another “best” value obtained so far by any
particle in the population, and this best value is a global best and called g
Best. After finding the two best values, the particle updates
its velocity and positions with following formulas (1) and (2): (1)
v
i
d
(
k
+
1
)
=
w
v
i
d
(
k
)
+
c
1
r
1
(
p
i
d
(
k
)
-
x
i
d
(
k
)
)
v
i
d
(
k
+
1
)
=
+
c
2
r
2
(
p
g
d
(
k
)
-
x
i
d
(
k
)
)
,
(2)
x
i
d
(
k
+
1
)
=
x
i
d
(
k
)
+
v
i
d
(
k
+
1
)
(
i
=
1,2
,
…
,
m
;
d
=
1,2
,
…
,
D
)
. In (1), P
i
=
(
p
i
1
,
p
i
2
,
…
,
p
i
D
) is the best previous position of the ith particle (also known as p
Best). According to the different definitions, there are two
different versions of PSO, with local version, and particles only have
information of their own and their neighbors’ bests, rather than that of
the entire neighborhood. If P
g
=
(
p
1
,
p
2
,
…
,
p
D
) is the best position among all the particles in the swarm
(also known as g
Best), such a version is called the global version. If P
g is taken from some smaller number of adjacent
particles of the population (also known as l
Best), such a version is called the local version. In (1) and (2)k represents
the iterative number andD is dimension of particles, and range
of particles are all determined by the problem to be optimized. Variables c
1
,
c
2 are learning factors, usually c
1
=
c
2
=
2, which represent the weighting of the stochastic acceleration
terms that pull each particle towards p
Best and g
Best positions. Thus, adjustment of these constants changes the
amount of “tension” in the system. r
1
~
U
(
0,1
), r
2
~
U
(
0,1
), and w are an inertia weight, which is initialized
typically in the range of [
0,1
]. Inertia weight controls the impact of previous historical
values of particle velocity on its current one. A larger inertia weight
pressures towards global exploration (searching new area), while a smaller
inertia weight pressures toward fine-tuning the current search area.
Particle’s velocities on each dimension are confined to a maximum
velocity V
max
⁡
~
x which is a parameter specified by the user. If the iteration
formulas would cause the velocity on that dimension to exceed V
max
⁡
~
x, then the velocity on that dimension is limited to V
max
⁡
. The termination criterion for the iterations is determined
according to whether the max generation or a designated value of the fitness of P
g is reached.The main disadvantage of the above original PSO is that it is difficult to keep
the diversity of population and to balance local and global search and hence it
may result in local optimal solutions. Besides, their search rates are commonly
lower and sometimes need more computation when solving some difficult
optimization problems. The original PSO frequently gets into the local solution,
especially when the problem size is middle or large. In the following a local
and global search combine particle swarm optimization algorithm is proposed to
deal with the above disadvantages of original PSO in solving optimization
problems.


## 2.1. Description of Original Particle Swarm Optimization Algorithm

Particle swarm optimization (PSO) is an evolutionary computation technique, and
its concept originated as a simulation of a simplified social system. The major
difference is that the evolution computing techniques use genetic operators
whereas swarm intelligence techniques use the physical movements of the
individuals in the swarm. PSO is distinctly different from other
evolutionary-type methods in a way that it does not need complex encoding and
decoding processes and does not use the operation (such as crossover and/or
mutation), and it takes real numbers as particles in the aspects of
representation solution. From the procedure, one can learn that PSO shares many
common points with GA; for example, they start with a neighborhood of a randomly
generated population, evaluate the population with fitness values, update the
population and search for the optimum with random techniques, and do not
guarantee optimal. Similar to other population-based algorithms, PSO as an
optimization tool can solve a variety of difficult optimization problems.
Compared to GA, one of the important attractive factors of PSO is simple that
there are very few parameters to adjust. It can achieve the optimal or
near-optimal solutions in a rather short time without enormous iterative
computations in digital implementation.Swarm intelligence simulates a social behavior such as bird flocking to a
promising position for certain objectives in a multidimensional space. It is
initialized with a population of random solutions and searches for optima by
updating generations. PSO system combines local search method (through
self-experience) with global search methods (through neighboring experience),
attempting to balance exploration. Like evolutionary algorithm, PSO search uses
a population (called swarm) of individuals (called particles) that are updated
from iteration to iteration. Each particle represents a candidate position
(i.e., solution) to the problem at hand, resembling the chromosome of GA. A
particle is treated as a point in anM-dimension space, and the
status of a particle is characterized by its position and velocity. Initialized
with a swarm of random particles, PSO is achieved through particle flying along
the trajectory that will be adjusted based on the best experience or position of
the one particle (called local best) and ever found by all particles (called
global best). PSO updates a population of particles with the internal velocity
and attempts to profit from the discoveries of themselves and previous
experiences of all other companions.


## 2.2. The Model of Original Particle Swarm Optimization Algorithm

In every search iteration, each particle is updated by following two
“best” values. The first one is the best solution (fitness) it has
achieved so far, and this value is calledp
Best. Another “best” value obtained so far by any
particle in the population, and this best value is a global best and called g
Best. After finding the two best values, the particle updates
its velocity and positions with following formulas (1) and (2): (1)
v
i
d
(
k
+
1
)
=
w
v
i
d
(
k
)
+
c
1
r
1
(
p
i
d
(
k
)
-
x
i
d
(
k
)
)
v
i
d
(
k
+
1
)
=
+
c
2
r
2
(
p
g
d
(
k
)
-
x
i
d
(
k
)
)
,
(2)
x
i
d
(
k
+
1
)
=
x
i
d
(
k
)
+
v
i
d
(
k
+
1
)
(
i
=
1,2
,
…
,
m
;
d
=
1,2
,
…
,
D
)
. In (1), P
i
=
(
p
i
1
,
p
i
2
,
…
,
p
i
D
) is the best previous position of the ith particle (also known as p
Best). According to the different definitions, there are two
different versions of PSO, with local version, and particles only have
information of their own and their neighbors’ bests, rather than that of
the entire neighborhood. If P
g
=
(
p
1
,
p
2
,
…
,
p
D
) is the best position among all the particles in the swarm
(also known as g
Best), such a version is called the global version. If P
g is taken from some smaller number of adjacent
particles of the population (also known as l
Best), such a version is called the local version. In (1) and (2)k represents
the iterative number andD is dimension of particles, and range
of particles are all determined by the problem to be optimized. Variables c
1
,
c
2 are learning factors, usually c
1
=
c
2
=
2, which represent the weighting of the stochastic acceleration
terms that pull each particle towards p
Best and g
Best positions. Thus, adjustment of these constants changes the
amount of “tension” in the system. r
1
~
U
(
0,1
), r
2
~
U
(
0,1
), and w are an inertia weight, which is initialized
typically in the range of [
0,1
]. Inertia weight controls the impact of previous historical
values of particle velocity on its current one. A larger inertia weight
pressures towards global exploration (searching new area), while a smaller
inertia weight pressures toward fine-tuning the current search area.
Particle’s velocities on each dimension are confined to a maximum
velocity V
max
⁡
~
x which is a parameter specified by the user. If the iteration
formulas would cause the velocity on that dimension to exceed V
max
⁡
~
x, then the velocity on that dimension is limited to V
max
⁡
. The termination criterion for the iterations is determined
according to whether the max generation or a designated value of the fitness of P
g is reached.The main disadvantage of the above original PSO is that it is difficult to keep
the diversity of population and to balance local and global search and hence it
may result in local optimal solutions. Besides, their search rates are commonly
lower and sometimes need more computation when solving some difficult
optimization problems. The original PSO frequently gets into the local solution,
especially when the problem size is middle or large. In the following a local
and global search combine particle swarm optimization algorithm is proposed to
deal with the above disadvantages of original PSO in solving optimization
problems.


## 3. A Local and Global Search Combined Particle Swarm Optimization
Algorithm

### 3.1. The Model of a Local and Global Search Combined Particle Swarm
Optimization Algorithm

Because both local and global version have their own advantages and disadvantage,
respectively, one can use them both in the algorithm, while global version is
used to get quick result and local version can refine the search space. In
original PSO algorithm, the information of individual best and global best was
shared by next generation particles, in which it tends to get stuck in a near
optimal solution especially for middle and large size problems. A local and
global combine particle swarm optimization (LGSCPSO) algorithm is presented in
this paper, in which the particles of next generation share the best information
of one particle itself, the best of particles in neighborhood population (local
best particles), and the best of all particles in the swarm (global best
particles). Among the neighborhood population, the particles are constituted by
the best particles of every generation with different proportion, which is
calledl
Best. As the generation increases, neighborhood population
diversity will grow larger and larger. LGSCPSO algorithm improves upon the
original PSO variation to increase accuracy of solution without sacrificing the
speed of solution significantly, and its detailed information will be given in
following.Suppose that the searching space isD-dimensional andm particles form the colony. The ith particle represents aD-dimensional vector X
i
=
(
x
i
1
,
x
i
2
,
…
,
x
i
D
)
(
i
=
1,2
,
…
,
m
). It means that the ith particle locates at X
i in the searching space and the position of each
particle is a potential solution. We could calculate the particle’s
fitness by putting its position into a designated objective function. When the
fitness is lower, the corresponding X
i is “better.” The ith particle’s “flying”
velocity is also aD-dimensional vector, denoted as V
i
=
(
v
i
1
,
v
i
2
,
…
,
v
i
D
). Denote the best position of the ith particle as P
i
=
(
p
i
1
,
p
i
2
,
…
,
p
i
D
), the best position of the local as P
l
=
(
p
l
1
,
p
l
2
,
…
,
p
l
D
), and the best position of the global as P
g
=
(
p
1
,
p
2
,
…
,
p
D
), respectively. After finding the three best values, the
particle of LGSCPSO updates its velocity and positions with the following
formulas: (3)
v
i
d
(
k
+
1
)
=
w
v
i
d
(
k
)
+
c
1
r
1
(
α
*
(
p
i
d
(
k
)
-
x
i
d
(
k
)
)
hhhhhhhhhhh
+
(
1
-
α
)
hhhhhhhhhhh
*
(
p
l
d
(
k
)
-
x
i
d
(
k
)
)
)
+
c
2
r
2
(
p
g
(
k
)
-
x
i
d
(
k
)
)
,
(4)
x
i
d
(
k
+
1
)
=
x
i
d
(
k
)
+
x
i
d
(
k
+
1
)
, where w is an inertia weight, which is initialized
typically in the range of [
0,1
]. The neighborhood particles of k generation are composed by the best
neighborhood particle (local the particle) of k generation. α
∈
[
0,1
] is weight index that is chosen according to different
optimization problem, which reflects relatively important degree of the best
position of the ith particle and the best position of the kth generation neighborhood particle. The
variables c
1
,
c
2 are acceleration constant, which control how far a particle
will move in a single iteration, and others parameters are same as in (1) and (2).The searching is a repeat process, and the stop criteria are that the maximum
iteration number is reached or the minimum error condition is satisfied. The
stop condition depends on the problem to be optimized. In LGSCPSO algorithm,
each particle of the swarm shares mutual information globally and benefits from
discoveries and previous experiences of all other colleagues during the search
process.


### 3.2. Pseudocode of LGSCPSO (See Pseudocode1)

Because PSO has very deep intelligent background, it is suitable for science
computation and engineering applications. A thorough mathematical foundation for
the methodology was not developed so far with the step of the algorithm;
however, it has been proven to be very effective for application.Pseudocode 1
Step  1: Let parameters; including swarm
size PS, maximum of generation endgen
and others
parameters will be used in LGSCPSO
algorithm.
Step  2: Iteration process
(i) Generate stochastically initialization population and
velocity;
(ii) Evaluate each particle’s fitness;
(iii) Initializeg
B
e
s
t position with the lowest fitness particle in the whole
swarm;
(iv) Initializep
B
e
s
t position with a copy of particle itself;
(v) Initializel
B
e
s
t position with the best particle of initializing
population;
(vi)k
≔ 0;
While (the maximum endgen of generation is
not met)
{
(i) k
≔
k
+
1;
(ii) Generate next swarm by (3), (4);
(iii) Evaluate swarm;
{ (a) Compute each particle’s fitness
in the swarm;
(b) Find newP
g, P
l of the swarm and P
i of each particle by comparison, and
update P
i and P
g;
(c) UpdateP
l using the best particle of k generation colony;}
}
Step  3: Output optimization results.


### 3.3. Convergence Analysis of LGSCPSO

In this section the LGSCPSO algorithm’s convergence is studied.Theorem 1.
In LGSCPSO algorithm in which its recurrence equation is (3) and (4), when parameters relation c
1
r
1
+
c
2
r
2
≤
1
+
w
-
2
w or 1
+
w
+
2
w
≤
c
1
r
1
+
c
2
r
2
<
2
w
+
2 is satisfied, it is convergent.Proof.
LetR
1
=
c
1
r
1 and R
2
=
c
2
r
2; from (3) and (4) we have (5)
x
i
d
(
k
+
1
)
=
(
1
-
R
1
-
R
2
)
x
i
d
(
k
)
+
w
v
i
d
(
k
)
+
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
. From (4) we can get v
i
d
(
k
)
=
x
i
d
(
k
)
-
x
i
d
(
k
-
1
), and combining (5) we have (6)
x
i
d
(
k
+
1
)
-
(
1
+
w
-
R
1
-
R
2
)
x
i
d
(
k
)
+
w
x
i
d
(
k
-
1
)
=
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
. LGSCPSO algorithm’s recurrence
equation is linearity constant coefficient nonhomogeneous, and the secular
equation of corresponding homogeneous recurrence equation is as follows: (7)
x
2
-
(
1
+
w
-
R
1
-
R
2
)
x
+
w
=
0
. Latent roots of above secular equation are as
follows: (8)
μ
1
=
1
+
w
-
R
1
-
R
2
+
(
1
+
w
-
R
1
-
R
2
)
2
-
4
w
2
,
μ
2
=
1
+
w
-
R
1
-
R
2
-
(
1
+
w
-
R
1
-
R
2
)
2
-
4
w
2
.
According to the relations of recurrence equation and its special solution,
let the special solution of LGSCPSO algorithm’s recurrence equation
(6) H
*
(
k
)
=
S substitute into (6), and we can solve (9)
S
=
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
R
1
+
R
2
.
According to the relations of recurrence equation’s general solution,
special solution, and its latent roots, we can obtain the general solution
of LGSCPSO algorithm’s recurrence equation (6) as follows: (10)
x
(
k
)
=
S
+
c
1
′′
μ
1
k
+
c
2
′′
μ
2
k
.
The initial solution and velocity of LGSCPSO algorithm recurrence equation is
stochastically generated; suppose first and second solution isx
i
d
(
1
) and x
i
d
(
2
), respectively, and according to (10), we can obtain that (11)
c
1
′′
=
μ
2
(
S
-
x
(
1
)
)
+
x
(
2
)
-
S
μ
1
(
μ
1
-
μ
2
)
,
c
2
′′
=
μ
1
(
x
(
1
)
-
S
)
-
x
(
2
)
+
S
μ
2
(
μ
1
-
μ
2
)
. To use (11) substitution into (10) simplification, we can
obtain the general solution of LGSCPSO recurrence equation (6) as follows: (12)
x
(
k
)
=
S
+
μ
2
(
S
-
x
(
1
)
)
+
x
(
2
)
-
S
μ
1
-
μ
2
*
μ
1
k
-
1
+
μ
1
(
x
(
1
)
-
S
)
-
x
(
2
)
+
S
μ
1
-
μ
2
*
μ
2
k
-
1
. If the limit of above x
(
k
) exists when k
→
∞, the LGSCPSO algorithm is convergent.
Let(13)
f
(
x
)
=
x
2
-
(
1
+
w
-
R
1
-
R
2
)
x
+
w
. So the Δ of above function is as follows: (14)
Δ
=
(
1
+
w
-
R
1
-
R
2
)
2
-
4
w
. Evidently, when (15)
f
(
1
)
>
0
,
f
(
-
1
)
>
0
,
Δ
≥
0 hold, we can obtain the function (13) latent roots -
1
<
μ
1
,
μ
2
<
1.
Namely, when0
<
c
1
r
1
+
c
2
r
2
≤
1
+
w
-
2
w or 1
+
w
+
2
w
≤
c
1
r
1
+
c
2
r
2
<
2
w
+
2 is satisfied, we can obtain -
1
<
μ
1
,
μ
2
<
1.
From LGSCPSO algorithm theory, we obviously get thatp
i
d
(
k
) and p
g
d
(
k
) are descending with k increase, and S is also descending with k increase.
When0
<
c
1
r
1
+
c
2
r
2
≤
1
+
w
-
2
w or 1
+
w
+
2
w
≤
c
1
r
1
+
c
2
r
2
<
2
w
+
2 is satisfied, namely, -
1
<
μ
1
,
μ
2
<
1 hold, we have (16)
lim
⁡
k
→
∞
x
(
k
)
=
lim
⁡
k
→
∞
(
S
+
μ
2
(
S
-
x
(
1
)
)
+
x
(
2
)
-
S
μ
1
-
μ
2
*
μ
1
k
-
1
hhhhhhh
+
μ
1
(
x
(
1
)
-
S
)
-
x
(
2
)
+
S
μ
1
-
μ
2
μ
2
k
-
1
)
=
lim
⁡
k
→
∞
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
R
1
+
R
2
. Because (
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
)
/
(
R
1
+
R
2
) is monotone decreasing and have lower bound
which is its optimum solution, lim
⁡
k
→
∞
(
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
)
/
(
R
1
+
R
2
) exist, namely, the endless particle
sequence brought about by LGSCPSO algorithm’s recurrence equation is
limited. So we can obtain that the LGSCPSO algorithm is convergent when its
parameters relations 0
<
c
1
r
1
+
c
2
r
2
≤
1
+
w
-
2
w or 1
+
w
+
2
w
≤
c
1
r
1
+
c
2
r
2
<
2
w
+
2 is satisfied.


## 3.1. The Model of a Local and Global Search Combined Particle Swarm
Optimization Algorithm

Because both local and global version have their own advantages and disadvantage,
respectively, one can use them both in the algorithm, while global version is
used to get quick result and local version can refine the search space. In
original PSO algorithm, the information of individual best and global best was
shared by next generation particles, in which it tends to get stuck in a near
optimal solution especially for middle and large size problems. A local and
global combine particle swarm optimization (LGSCPSO) algorithm is presented in
this paper, in which the particles of next generation share the best information
of one particle itself, the best of particles in neighborhood population (local
best particles), and the best of all particles in the swarm (global best
particles). Among the neighborhood population, the particles are constituted by
the best particles of every generation with different proportion, which is
calledl
Best. As the generation increases, neighborhood population
diversity will grow larger and larger. LGSCPSO algorithm improves upon the
original PSO variation to increase accuracy of solution without sacrificing the
speed of solution significantly, and its detailed information will be given in
following.Suppose that the searching space isD-dimensional andm particles form the colony. The ith particle represents aD-dimensional vector X
i
=
(
x
i
1
,
x
i
2
,
…
,
x
i
D
)
(
i
=
1,2
,
…
,
m
). It means that the ith particle locates at X
i in the searching space and the position of each
particle is a potential solution. We could calculate the particle’s
fitness by putting its position into a designated objective function. When the
fitness is lower, the corresponding X
i is “better.” The ith particle’s “flying”
velocity is also aD-dimensional vector, denoted as V
i
=
(
v
i
1
,
v
i
2
,
…
,
v
i
D
). Denote the best position of the ith particle as P
i
=
(
p
i
1
,
p
i
2
,
…
,
p
i
D
), the best position of the local as P
l
=
(
p
l
1
,
p
l
2
,
…
,
p
l
D
), and the best position of the global as P
g
=
(
p
1
,
p
2
,
…
,
p
D
), respectively. After finding the three best values, the
particle of LGSCPSO updates its velocity and positions with the following
formulas: (3)
v
i
d
(
k
+
1
)
=
w
v
i
d
(
k
)
+
c
1
r
1
(
α
*
(
p
i
d
(
k
)
-
x
i
d
(
k
)
)
hhhhhhhhhhh
+
(
1
-
α
)
hhhhhhhhhhh
*
(
p
l
d
(
k
)
-
x
i
d
(
k
)
)
)
+
c
2
r
2
(
p
g
(
k
)
-
x
i
d
(
k
)
)
,
(4)
x
i
d
(
k
+
1
)
=
x
i
d
(
k
)
+
x
i
d
(
k
+
1
)
, where w is an inertia weight, which is initialized
typically in the range of [
0,1
]. The neighborhood particles of k generation are composed by the best
neighborhood particle (local the particle) of k generation. α
∈
[
0,1
] is weight index that is chosen according to different
optimization problem, which reflects relatively important degree of the best
position of the ith particle and the best position of the kth generation neighborhood particle. The
variables c
1
,
c
2 are acceleration constant, which control how far a particle
will move in a single iteration, and others parameters are same as in (1) and (2).The searching is a repeat process, and the stop criteria are that the maximum
iteration number is reached or the minimum error condition is satisfied. The
stop condition depends on the problem to be optimized. In LGSCPSO algorithm,
each particle of the swarm shares mutual information globally and benefits from
discoveries and previous experiences of all other colleagues during the search
process.


## 3.2. Pseudocode of LGSCPSO (See Pseudocode1)

Because PSO has very deep intelligent background, it is suitable for science
computation and engineering applications. A thorough mathematical foundation for
the methodology was not developed so far with the step of the algorithm;
however, it has been proven to be very effective for application.Pseudocode 1
Step  1: Let parameters; including swarm
size PS, maximum of generation endgen
and others
parameters will be used in LGSCPSO
algorithm.
Step  2: Iteration process
(i) Generate stochastically initialization population and
velocity;
(ii) Evaluate each particle’s fitness;
(iii) Initializeg
B
e
s
t position with the lowest fitness particle in the whole
swarm;
(iv) Initializep
B
e
s
t position with a copy of particle itself;
(v) Initializel
B
e
s
t position with the best particle of initializing
population;
(vi)k
≔ 0;
While (the maximum endgen of generation is
not met)
{
(i) k
≔
k
+
1;
(ii) Generate next swarm by (3), (4);
(iii) Evaluate swarm;
{ (a) Compute each particle’s fitness
in the swarm;
(b) Find newP
g, P
l of the swarm and P
i of each particle by comparison, and
update P
i and P
g;
(c) UpdateP
l using the best particle of k generation colony;}
}
Step  3: Output optimization results.


## 3.3. Convergence Analysis of LGSCPSO

In this section the LGSCPSO algorithm’s convergence is studied.Theorem 1.
In LGSCPSO algorithm in which its recurrence equation is (3) and (4), when parameters relation c
1
r
1
+
c
2
r
2
≤
1
+
w
-
2
w or 1
+
w
+
2
w
≤
c
1
r
1
+
c
2
r
2
<
2
w
+
2 is satisfied, it is convergent.Proof.
LetR
1
=
c
1
r
1 and R
2
=
c
2
r
2; from (3) and (4) we have (5)
x
i
d
(
k
+
1
)
=
(
1
-
R
1
-
R
2
)
x
i
d
(
k
)
+
w
v
i
d
(
k
)
+
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
. From (4) we can get v
i
d
(
k
)
=
x
i
d
(
k
)
-
x
i
d
(
k
-
1
), and combining (5) we have (6)
x
i
d
(
k
+
1
)
-
(
1
+
w
-
R
1
-
R
2
)
x
i
d
(
k
)
+
w
x
i
d
(
k
-
1
)
=
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
. LGSCPSO algorithm’s recurrence
equation is linearity constant coefficient nonhomogeneous, and the secular
equation of corresponding homogeneous recurrence equation is as follows: (7)
x
2
-
(
1
+
w
-
R
1
-
R
2
)
x
+
w
=
0
. Latent roots of above secular equation are as
follows: (8)
μ
1
=
1
+
w
-
R
1
-
R
2
+
(
1
+
w
-
R
1
-
R
2
)
2
-
4
w
2
,
μ
2
=
1
+
w
-
R
1
-
R
2
-
(
1
+
w
-
R
1
-
R
2
)
2
-
4
w
2
.
According to the relations of recurrence equation and its special solution,
let the special solution of LGSCPSO algorithm’s recurrence equation
(6) H
*
(
k
)
=
S substitute into (6), and we can solve (9)
S
=
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
R
1
+
R
2
.
According to the relations of recurrence equation’s general solution,
special solution, and its latent roots, we can obtain the general solution
of LGSCPSO algorithm’s recurrence equation (6) as follows: (10)
x
(
k
)
=
S
+
c
1
′′
μ
1
k
+
c
2
′′
μ
2
k
.
The initial solution and velocity of LGSCPSO algorithm recurrence equation is
stochastically generated; suppose first and second solution isx
i
d
(
1
) and x
i
d
(
2
), respectively, and according to (10), we can obtain that (11)
c
1
′′
=
μ
2
(
S
-
x
(
1
)
)
+
x
(
2
)
-
S
μ
1
(
μ
1
-
μ
2
)
,
c
2
′′
=
μ
1
(
x
(
1
)
-
S
)
-
x
(
2
)
+
S
μ
2
(
μ
1
-
μ
2
)
. To use (11) substitution into (10) simplification, we can
obtain the general solution of LGSCPSO recurrence equation (6) as follows: (12)
x
(
k
)
=
S
+
μ
2
(
S
-
x
(
1
)
)
+
x
(
2
)
-
S
μ
1
-
μ
2
*
μ
1
k
-
1
+
μ
1
(
x
(
1
)
-
S
)
-
x
(
2
)
+
S
μ
1
-
μ
2
*
μ
2
k
-
1
. If the limit of above x
(
k
) exists when k
→
∞, the LGSCPSO algorithm is convergent.
Let(13)
f
(
x
)
=
x
2
-
(
1
+
w
-
R
1
-
R
2
)
x
+
w
. So the Δ of above function is as follows: (14)
Δ
=
(
1
+
w
-
R
1
-
R
2
)
2
-
4
w
. Evidently, when (15)
f
(
1
)
>
0
,
f
(
-
1
)
>
0
,
Δ
≥
0 hold, we can obtain the function (13) latent roots -
1
<
μ
1
,
μ
2
<
1.
Namely, when0
<
c
1
r
1
+
c
2
r
2
≤
1
+
w
-
2
w or 1
+
w
+
2
w
≤
c
1
r
1
+
c
2
r
2
<
2
w
+
2 is satisfied, we can obtain -
1
<
μ
1
,
μ
2
<
1.
From LGSCPSO algorithm theory, we obviously get thatp
i
d
(
k
) and p
g
d
(
k
) are descending with k increase, and S is also descending with k increase.
When0
<
c
1
r
1
+
c
2
r
2
≤
1
+
w
-
2
w or 1
+
w
+
2
w
≤
c
1
r
1
+
c
2
r
2
<
2
w
+
2 is satisfied, namely, -
1
<
μ
1
,
μ
2
<
1 hold, we have (16)
lim
⁡
k
→
∞
x
(
k
)
=
lim
⁡
k
→
∞
(
S
+
μ
2
(
S
-
x
(
1
)
)
+
x
(
2
)
-
S
μ
1
-
μ
2
*
μ
1
k
-
1
hhhhhhh
+
μ
1
(
x
(
1
)
-
S
)
-
x
(
2
)
+
S
μ
1
-
μ
2
μ
2
k
-
1
)
=
lim
⁡
k
→
∞
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
R
1
+
R
2
. Because (
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
)
/
(
R
1
+
R
2
) is monotone decreasing and have lower bound
which is its optimum solution, lim
⁡
k
→
∞
(
R
1
α
p
i
d
(
k
)
+
R
1
(
1
-
α
)
p
l
d
(
k
)
+
R
2
p
g
d
(
k
)
)
/
(
R
1
+
R
2
) exist, namely, the endless particle
sequence brought about by LGSCPSO algorithm’s recurrence equation is
limited. So we can obtain that the LGSCPSO algorithm is convergent when its
parameters relations 0
<
c
1
r
1
+
c
2
r
2
≤
1
+
w
-
2
w or 1
+
w
+
2
w
≤
c
1
r
1
+
c
2
r
2
<
2
w
+
2 is satisfied.


## 4. Numerical Simulation

### 4.1. Test Functions

To illustrate the effectiveness and performance of LGSCPSO algorithm for
optimization problems, a set of 8 representative benchmark functions with
different dimensions were employed to evaluate it in comparison with original
PSO. Many authors tested algorithm using them widely.Sphere Model. Consider f
1
(
x
)
=
∑
i
=
1
n
x
i
2, where x
i
∈
[
-
100,100
].Schwefel’s Problem  1.2. Consider  f
2
(
x
)
=
∑
i
=
1
n
(
∑
j
=
1
i
x
j
)
2, where x
i
∈
[
-
100,100
].Schwefel’s Problem  2.21.  Consider f
3
(
x
)
=
max
⁡
i
{
|
x
i
|
,
1
≤
i
≤
n
}, where x
i
∈
[
-
100,100
].Generalized Rosenbrock’s Function. Consider f
4
(
x
)
=
∑
i
=
1
n
-
1
(
100
(
x
i
+
1
-
x
i
2
)
2
+
(
x
i
-
1
)
2
), where x
i
∈
[
-
100,100
].Ackley’s Function. Consider f
5
(
x
)
=
-
20
exp
⁡
(
-
0.2
(
1
/
n
)
∑
i
=
1
n
x
i
2
)
-
exp
⁡
(
(
1
/
n
)
∑
i
=
1
n
cos
⁡
2
π
x
i
)
+
20
+
e, where x
i
∈
[
-
32,32
].Generalized Griewank Function. Consider f
6
(
x
)
=
(
1
/
4000
)
∑
i
=
1
n
(
x
i
-
100
)
2
-
∏
i
=
1
n
cos
⁡
(
(
x
i
-
100
)
/
i
)
+
1, where x
i
∈
[
-
600,600
].Generalized Penalized Functions. Consider (17)
f
7
(
x
)
=
π
n
{
10
si
n
2
(
π
y
1
)
+
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
h
h
h
×
[
1
+
10
si
n
2
(
π
y
i
+
1
)
]
+
(
y
n
-
1
)
2
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
}
+
∑
i
=
1
n
u
(
x
i
,
10,100,4
)
,
f
8
(
x
)
=
0.1
{
si
n
2
(
π
y
1
)
+
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
h
h
h
h
×
[
1
+
10
si
n
2
(
π
y
i
+
1
)
]
+
(
y
n
-
1
)
2
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
}
+
∑
i
=
1
n
u
(
x
i
,
10,100,4
)
, where (18)
u
(
x
i
,
a
,
k
,
m
)
=
{
k
(
x
i
-
a
)
m
,
x
i
>
a
0
,
-
a
≤
x
i
≤
a
k
(
-
x
i
-
a
)
m
,
x
i
<
-
a
,
y
i
=
1
+
(
1
4
)
(
x
i
+
1
)
,
x
i
∈
[
-
50,50
]
.They can be divided into unimodal (functionf
1
–
f
3) and multimodal functions (function f
4
–
f
8), where the number of local minima increases exponentially
with the problem dimension.


### 4.2. Experimental Results and Comparison

The experimental results of original PSO and LGSCPSO algorithm on each test
function are listed in Table1. To get
the average performance of the LGSCPSO algorithm ten runs on each problem
instance were performed and the solution quality was averaged. The comparisons
results of LGSCPSO and Original PSO algorithm are shown in Table 1 of the appendix.Table 1
The comparison results of the PSO algorithm and the LGSCPSO
algorithm.
Min./Max./Average (e
~
n represents ×
10
n)
a
w
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
1,
Dim 150,
Best 0;
PS 150,
EG 2500
.1.2.3.4.5.6.7
.8 .9
7.9247/798.1366/119.27351.0216/328.5215/36.36210.1788/3.6762/0.95800.0276/1.0302/0.17320.0015/0.2004/0.0345
6.199e  −  4/20.4397/2.1839 8.7194e  −  4/32.076/3.51180.0027/1.0443/0.37940.0301/3.1973e  +  5/5.7818e  +  4
4.8684/788.0086/137.77270.2372/39.35/5.29250.0516/72.579/18.80860.0024/16.9442/2.0285
0.0011/6.0067/0.7585 0.1109/326.9967/38.2332217.0564/2.5137e  +  3/922.9063131.8696/1.0093e  +  3/605.76083.5023/3.5197e  +  5/7.5222e  +  4
3.4298/466.135/129.40390.2165/36.7093/9.15460.0446/0.9215/0.32450.0081/5.1882/0.60590.0021/15.3744/3.103
9.091e  −  4/50.1812/9.9065 0.1802/416.0314/88.517622.395/2.7571e  +  3/760.493.3725/3.3151e  +  5/3.3313e  +  4
6.2134/108.3578/34.96780.336/297.5513/35.94330.0931/2.9157/0.69560.0255/12.8562/1.50620.0042/1.563/0.2760.0031/1.3039/0.21721.5877e  −  4/2.5115/0.3145
4.9801e  −  5/0.0052/8.8123e  −  4*9.7207e  −  5/390.2679/40.2488
202.8074/3.8168e  +  3/899.134725.8671/126.0089/74.35552.0381/42.8/14.73930.2738/12.3747/2.66930.0354/0.9538/0.21240.0088/0.3742/0.0659
0.0027/0.3203/0.05540.0104/0.1721/0.06780.5088/4.2984e  +  5/2.6068e  +  5
PSO
3.5545e  +  5/4.2006e  +  5/3.9996e  +  5
3.6563e  +  5/4.1904e  +  5/3.9654e  +  5
3.8295e  +  5/4.1465e  +  5/4.0243e  +  5
0.5082/4.185e  +  5/3.4273e  +  5
3.834e  +  5/4.2439e  +  5/4.0552e  +  5
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
2,
Dim 30,
Best 0;
PS 150,
EG 2000
.1.2.3
.4 .5.6.7.8.9
1.5314e  −  5/0.0012/2.3471e  −  47.544e  −  6/4.1057e  −  4/1.2064e  −  4
3.3468e  −  6/2.5205e  −  4/8.7442e  −  52.2108e  −  5/3.3094e  −  4/9.2148e  −  51.4607e  −  5/3.3855e  −  4/9.0081e  −  51.4749e  −  4/0.0097/0.00162.2105e  −  4/0.0705/0.02320.0879/0.9873/0.47321.3288/15.724/5.0923
3.4819e  −  8/9.7461e  −  7/2.8081e  −  73.6274e  −  9/2.252e  −  7/6.7068e  −  81.7849e  −  10/6.8322e  −  8/1.3459e  −  89.6468e  −  11/1.3930e  −  7/2.0447e  −  8
7.8029e  −  11/9.93e  −  9/4.5009e  −  93.0876e  −  9/4.2278e  −  7/5.6384e  −  81.8076e  −  7/1.0146e  −  5/2.1222e  −  61.437e  −  6/6.4389e  −  4/2.8104e  −  40.0042/0.1077/0.0241
1.6392e  −  9/8.0260e  −  8/2.4483e  −  84.3649e  −  11/1.9808e  −  8/3.0528e  −  94.3800e  −  11/8.3452e  −  9/2.5903e  −  9
2.9752e  −  12/5.6342e  −  10/1.6065e  −  10* 7.1437e  −  12/4.1282e  −  9/8.6320e  −  102.0796e  −  11/5.0013e  −  9/1.0295e  −  91.1123e  −  10/2.6655e  −  8/8.6717e  −  98.3474e  −  8/7.0419e  −  6/2.2156e  −  67.8655e  −  5/0.0055/0.002
1.9438e  −  8/4.5893e  −  6/1.0199e  −  66.9751e  −  9/4.4344e  −  6/5.5416e  −  73.0993e  −  10/2.9669e  −  8/6.9531e  −  9
5.1336e  −  11/6.7099e  −  8/9.4611e  −  91.2011e  −  10/6.3320e  −  9/1.7701e  −  94.0553e  −  10/3.5928e  −  8/5.5263e  −  96.2429e  −  9/6.8153e  −  7/1.8691e  −  79.0305e  −  7/1.0765e  −  4/1.9495e  −  52.6371e  −  4/0.0057/0.0024
0.0016/0.0746/0.01814.1506e  −  5/0.002/5.1297e  −  41.3450e  −  5/1.3095e  −  4/6.5852e  −  51.1121e  −  6/2.0099e  −  5/6.5348e  −  61.5491e  −  6/1.6945e  −  5/6.7154e  −  6
3.3047e  −  7/7.1073e  −  5/2.0406e  −  52.4160e  −  5/5.8853e  −  4/1.5538e  −  46.3392e  −  4/0.0096/0.00420.0595/1.9094/0.3682
PSO
364.253/2.4939e  +  4/1.0134e  +  4
1.3765/4.6947e  +  3/481.8464
0.0691/2.5876/0.7283
0.1723/3.9289/1.8005
17.3828/984.8965/188.1513
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
3,
Dim 30,
Best 0;
PS 150,
EG 2000
.1.2.3
.4 .5.6.7.8.9
2.3621e  −  4/0.0013/7.311e  −  46.0977e  −  5/4.2674e  −  4/2.1957e  −  45.5341e  −  5/7.0551e  −  4/2.1313e  −  4
4.9251e  −  5/3.4147e  −  4/1.4638e  −  45.2814e  −  5/0.0019/4.5195e  −  44.8841e  −  4/0.0053/0.00290.0046/0.0476/0.01910.0569/0.5819/0.22660.4589/4.2785/1.3995
1.3902e  −  5/2.0080e  −  4/8.8198e  −  51.1701e  −  6/2.8489e  −  5/8.0878e  −  63.8603e  −  7/8.6707e  −  6/1.7948e  −  6
1.4e  −  7/5.6791e  −  6/1.3055e  −  61.6044e  −  7/1.6259e  −  6/5.7405e  −  71.7168e  −  7/6.7027e  −  6/2.6332e  −  69.6091e  −  6/7.5094e  −  5/2.8845e  −  58.2486e  −  5/0.0012/3.9755e  −  40.001/0.0436/0.0158
1.6884e  −  5/3.3877e  −  4/1.194e  −  47.4212e  −  7/4.1010e  −  6/1.8869e  −  65.1647e  −  8/1.5398e  −  6/3.7869e  −  7
1.5353e  −  9/1.8722e  −  7/6.9333e  −  8* 4.7963e  −  9/3.1684e  −  7/6.3918e  −  82.0730e  −  9/6.1577e  −  7/9.0325e  −  83.4465e  −  8/7.9999e  −  6/1.0596e  −  61.1441e  −  6/1.3502e  −  5/5.7652e  −  62.4963e  −  5/0.0027/6.1308e  −  4
1.5547e  −  5/6.6884e  −  4/1.4984e  −  43.5838e  −  7/8.3143e  −  6/4.7977e  −  63.9851e  −  8/2.4747e  −  6/7.3211e  −  73.4271e  −  8/4.2062e  −  7/1.0447e  −  76.6274e  −  9/8.7745e  −  7/1.4709e  −  7
2.0358e  −  9/4.6960e  −  8/1.9028e  −  83.9048e  −  8/3.9274e  −  7/1.5350e  −  77.5274e  −  7/1.2135e  −  5/3.6172e  −  64.4217e  −  5/5.6209e  −  4/1.7621e  −  4
0.001/0.0262/0.00551.1206e  −  5/5.1122e  −  4/1.9353e  −  43.1555e  −  6/3.1428e  −  5/1.2056e  −  51.8118e  −  7/1.3756e  −  5/3.7208e  −  62.2951e  −  7/1.1701e  −  6/5.3581e  −  7
8.2804e  −  8/2.1413e  −  5/3.1923e  −  61.0194e  −  6/1.0954e  −  5/3.7688e  −  69.2958e  −  6/1.813e  −  4/1.0582e  −  40.0016/0.0311/0.0073
PSO
0.9442/7.2642/3.7178
0.1348/0.5193/0.2436
0.005/0.0675/0.0212
0.0084/0.043/0.03
0.2007/0.9216/0.4999
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
4,
Dim 25,
Best 0;
PS 150,
EG 2500
.1.2.3
.4 .5.6.7.8.9
1.8571/1.229e  +  3/135.88070.1568/2.5441e  +  3/388.20910.6066/799.8443/169.64220.2485/181.0596/25.4415
0.0237/40.8918/11.14410.0253/276.2887/57.01080.0916/228.4057/30.79650.4745/435.3273/56.96344.3957/341.7015/67.9295
6.2873/3.2863e  +  3/709.82220.0113/789.7377/115.4752.3034/796.4533/123.7322
9.6966e  −  5/6.8529/2.0791* 0.0069/23.873/4.99742.547e  −  4/9.0536/4.0420.0702/14.2596/6.66970.0189/16.2452/9.93793.2415/838.6487/128.4819
0.2013/285.8471/44.284.0693/4.9375e  +  3/562.5381
3.3983e  −  4/302.2135/32.97580.2062/179.2151/21.4179
3.5267e  −  4/3.8396/0.84943.6341e  −  4/6.6633/3.65670.0372/152.4195/22.09720.1941/238.9312/38.12250.7361/334.4369/49.1472
0.5944/144.3777/23.71881.8021/937.5994/153.76820.0455/207.5395/40.31320.0121/42.4684/5.2882
0.0018/182.335/21.08311.4713/10.0475/5.65290.1549/12.1124/7.78820.5691/1.2973e  +  3/170.2970.2681/312.4161/40.7414
0.0678/486.002/165.06634.3764/7.2066e  +  3/1.0582e  +  30.063/197.4164/32.77493.4459/962.925/115.1339
0.0125/423.4865/75.33540.1299/26.4581/10.28620.1671/672.3137/106.70180.6944/213.5931/30.54170.2055/1.1812e  +  3/161.6274
PSO
4.3221e  +  4/5.1189e  +  4/4.6655e  +  4
3.9971e  +  4/5.4303e  +  4/4.6593e  +  4
2.876e  +  4/5.5191e  +  4/4.7462e  +  4
3.937e  +  4/5.2589e  +  4/4.6619e  +  4
8.4709/102.7603/23.237
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
5,
Dim 30
Best 0;
PS 100,
EG 2000
.1.2
.3 .4.5.6.7.8.9
7.9936e  −  15/4.3521e  −  14/2.256e  −  147.9936e  −  15/7.9936e  −  15/7.9936e  −  154.4409e  −  15/7.9936e  −  15/7.2831e  −  15
4.4409e  −  15/7.9936e  −  15/5.1514e  −  154.4409e  −  15/7.9936e  −  15/5.8620e  −  154.4409e  −  15/2.3162/0.23164.4409e  −  15/1.1908e  −  9/1.1908e  −  106.9209e  −  5/2.2201/0.73284.8998e  −  5/1.6462/0.3991
7.9936e  −  15/7.9936e  −  15/7.9936e  −  154.4409e  −  15/7.9936e  −  15/5.1514e  −  15
4.4409e  −  15/7.9936e  −  15/5.5067e  −  15* 4.4409e  −  15/1.6462/0.28014.4409e  −  15/2.3162/0.59674.4409e  −  15/2.3162/1.09347.9936e  −  15/2.887/1.10890.0021/3.0268/1.75710.0043/2.5791/1.7115
4.4409e  −  15/7.9936e  −  15/7.2831e  −  15
4.4409e  −  15/7.9936e  −  15/5.1514e  −  154.4409e  −  15/1.5017/0.15024.4409e  −  15/2.0119/0.31674.4409e  −  15/1.6462/0.71284.4409e  −  15/3.6807/1.35854.4409e  −  15/2.4949/1.25647.9936e  −  15/2.0133/1.23727.9936e  −  15/2.6602/0.8856
7.9936e  −  15/1.5099e  −  14/9.4147e  −  154.4409e  −  15/7.9936e  −  15/7.2831e  −  15
4.4409e  −15/7.9936e  −  15/5.5067e  −  15* 4.4409e  −  15/2.5795/0.37354.4409e  −  15/2.0133/0.54.4409e  −  15/1.3404/0.22724.4409e  −  15/1.3404/0.1344.4409e  −  15/2.0133/0.54.4409e  −  15/1.1551/0.1155
1.056e  −  11/2.0851e  −  10/7.4168e  −  111.8652e  −  14/4.1656e  −  13/9.1838e  −  147.9936e  −  15/1.5099e  −  14/8.7041e  −  15
4.4409e  −  15/7.9936e  −  15/5.862e  −  15
4.4409e  −  15/7.9936e  −  15/5.862e  −  15 4.4409e  −  15/7.9936e  −  15/6.9278e  −  154.4409e  −  15/7.9936e  −  15/7.6383e  −  154.4409e  −  15/7.9936e  −  15/7.6383e  −  157.9936e  −  15/7.9936e  −  15/7.9936e  −  15
PSO
2.4655e  −  5/2.2315/1.1133
1.6387e  −  4/2.8859/0.7401
7.9936e  −  15/2.4959/0.6931
4.4409e  −  15/7.9936e  −  15/7.2831e  −  15
7.9936e  −  15/1.5099e  −  14/1.2967e  −  14
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
6,
Dim 150,
Best 0;
PS 150,
EG 2500
.1.2.3.4.5.6.7.8
.9
0.9416/1.4223/1.16350.2941/0.9446/0.48420.0383/0.2148/0.09940.002/0.3372/0.09454.6926e  −  4/0.2611/0.0958
3.7473e  −  5/0.1647/0.05411.572e  −  4/0.0664/0.01941.1609e  −  4/0.2322/0.05410.0129/3.4498e  +  3/639.0998
0.5782/1.452/0.94520.0804/2.7225/0.50280.0036/0.8091/0.15716.6171e  −  4/1.0183/0.1775
8.1220e  −  5/2.1079/0.41950.2266/6.0141/2.12031.6879/24.077/11.22931.9272/17.0006/7.12860.2582/3.7637e  +  3/668.9854
0.3887/8.8918/1.72370.0397/2.1197/0.46020.0055/4.4603/0.55018.199e  −  4/0.4379/0.122
2.5102e  −  4/0.5375/0.31580.0023/3.2548/0.76760.2953/4.2413/1.16681.3151/17.8887/9.72290.0979/12.3986/3.8033
0.9863/3.3689/1.32780.3404/0.6563/0.48770.0084/0.2726/0.13320.0013/0.2891/0.13430.0044/0.9635/0.18523.0448e  −  4/1.2816/0.26870.0074/0.5053/0.14691.3352e  −  5/0.061/0.0226
5.0879e  −  6/1.1096/0.1934*
3.0738/12.5482/6.15131.1998/2.6794/1.50130.545/1.2007/0.82910.0217/6.2987/0.80920.0061/0.2465/0.06210.0015/0.3055/0.0567
2.6726e  −  4/0.1051/0.02414.8173e  −  4/0.0609/0.0190.0283/3.9697e  +  3/2.3201e  +  3
3.1058e  +  3/3.9496e  +  3/3.6383e  +  3
2.766e  +  3/3.952e  +  3/3.5857e  +  3
0.9187/3.9622e  +  3/2.9803e  +  3
0.1916/3.9592e  +  3/2.8647e  +  3
3.5377e  +  3/3.9083e  +  3/3.7889e  +  3
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
7,
Dim 100,
Best 0;
PS 150,
EG 2500
.1.2.3.4.5.6
.7 .8
0.0014/0.1015/0.02684.7654e  −  5/0.0311/0.00343.3537e  −  5/0.0628/0.0165
5.8595e  −  7/0.0311/0.00368.2125e  −  7/0.1685/0.0298
5.8595e  −  7/0.0311/0.00360.0022/2.5577/0.824630.7659/1.0297e  +  9/3.6691e  +  8
2.1663e  −  5/0.1245/0.01896.125e  −  7/0.0622/0.00977.1955e  −  9/0.0311/0.0062
4.1191e  −  9/0.1555/0.02495.3424e  −  8/0.3743/0.07174.1946e  −  8/1.731/0.37940.1557/2.8177/0.89953.3049e  −  4/2.7669e  +  8/2.7669e  +  7
1.0775e  −  5/0.1866/0.03142.6987e  −  7/0.1557/0.02499.1232e  −  9/0.1868/0.03742.9509e  −  9/0.1868/0.04675.2224e  −  7/0.6882/0.2318
5.6598e  −  10/1.3103/0.28724.3761e  −  5/1.3392/0.41911.3321e  −  5/1.4361/0.5313
3.9207e  −  4/0.3284/0.08883.7171e  −  6/0.2498/0.06053.4490e  −  7/0.1244/0.04673.8808e  −  9/0.499/0.08724.4126e  −  9/0.2497/0.06553.7644e  −  9/0.2808/0.0779
1.0263e  −  10/0.2490/0.0498* 8.0965e  −  6/0.2789/0.0448
0.8393/18.623/5.1570.2215/1.3457/0.6130.0018/0.5005/0.16016.0798e  −  4/0.3451/0.09477.0408e  −  6/0.137/0.0287
3.7725e  −  7/0.4987/0.14056.0553e  −  7/0.5295/0.08610.4766/3.2293e  +  3/326.8969
2.2171e  +  9/2.6638e  +  9/2.3786e  +  9
2.05e  +  9/2.6165e  +  9/2.2883e  +  9
1.9676e  +  9/2.8563e  +  9/2.3233e  +  9
1.3125e  +  9/2.6573e  +  9/2.1146e  +  9
1.8631e  +  9/2.8019e  +  9/2.2317e  +  9
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
8,
Dim 100,
Best 0;
PS 150,
EG 2500
.1.2.3.4.5
.6 .7.8
0.0014/0.1008/0.01872.0521e  −  4/0.0132/0.00399.8083e  −  5/0.9506/0.1062
3.0775e  −  7/0.0037/5.5843e  −  43.3459e  −  6/4.3813e  −  4/1.47e  −  42.891e  −  6/2.5064/0.4977.7979e  −  4/9.103/2.6232.0949e  +  9/2.7924e  +  9/2.4913e  +  9
6.9482e  −  5/0.1026/0.01148.9263e  −  7/8.6653e  −  5/2.0482e  −  52.1133e  −  7/4.1364e  −  5/9.8935e  −  6
1.8204e  −  8/5.2730e  −  5/1.0100e  −  54.9008e  −  8/6.3249e  −  4/8.0700e  −  51.1649e  −  7/1.8877/0.50092.1485e  −  4/2.7725/0.87362.1328e  +  /2.8009e  +  9/2.5344e  +  9
1.9512e  −  5/0.0023/7.8123e  −  41.3466e  −  6/0.2978/0.03072.7567e  −  8/1.1156e  −  4/2.2848e  −  5
8.9806e  −  10/1.4026e  −  4/2.3733e  −  51.4968e  −  9/0.5878/0.06884.0322e  −  7/3.0954/0.86182.6362e  −  6/6.9698/2.03451.816e  +  9/2.8674e  +  9/2.4126e  +  9
2.037e  −  4/0.1195/0.01836.6231e  −  5/0.0019/5.873e  −  41.1378e  −  6/0.0015/2.6495e  −  41.8884e  −  7/0.0905/0.00915.5391e  −  10/8.5755e  −  5/9.8527e  −  6
4.9993e  −  10/7.1912e  −  5/8.2796e  −  6* 3.5393e  −  9/0.0011/1.6126e  −  41.6523e  −  7/2.1877/0.3866
5.8424/86.3769/26.71520.4244/9.4505/2.60560.0034/3.0968/0.5817.1911e  −  5/1.4867/0.1498
7.1442e  −  7/0.0052/0.00171.0077e  −  6/4.2974e  −  4/7.1607e  −  51.0331e  −  5/0.2203/0.04861.2518/2.391e  +  9/2.391e  +  8
PSO
1.8688e  +  9/2.8661e  +  9/2.3999e  +  9
2.05e  +  9/2.6165e  +  9/2.2883e  +  9
1.5174e  +  9/2.7626e  +  9/2.2677e  +  9
1.3125e  +  9/2.6573e  +  9/2.1146e  +  9
1.8631e  +  9/2.8019e  +  9/2.2317e  +  9
Remark: In Table1,
Fun and Dim denote function
and its dimension, respectively; the Best is this
function’s optimum value; PS and
EG indicate the population size and algorithm
terminate generation. PSO and LGC denote the original PSO and
LGSCPSO algorithm. The better solutions and the parameters found in
LGSCPSO are illustrated with bold letters, and the best solutions
are shown in data with asterisk.From Table1 one can observe that w
∈
[
0.2,0.4
] have the highest performance since using them has smaller
minimum and smaller arithmetic mean in relation to the solutions obtained by the
others, especially w
≈
0.4 has the better search efficiency. In LGSCPSO algorithm,
various parameter α should be selected for optimizing different
problems, as a whole, it is better when α
∈
[
0.3
0.9
], especially α
≈
0.4. From simulation results we can obtain that the LGSCPSO
algorithm is clearly better than the original PSO for continuous nonlinear
function optimization problem. The convergence figures of most effective LGSCPSO
comparing with original PSO for 8 instances are as follows.From Figure1 we can discover that the
convergence rate of LGSCPSO algorithm is clearly faster than the OPSO on every
benchmark function, especially it is more efficacious than OPSO for middle and
large size optimization problem. Accordingly, we can state that the LGSCPSO
algorithm is more effective than OPSO algorithm. For test function 5, the search
capability of LGSCPSO algorithm is worse than that of OPSO at the beginning of
the iteration. However, seen from the enlarged figure, the search capability of
LGSCPSO is markedly enhanced along with the iteration increasing. In addition we
can find that the simulation data from test function 5 has a strong
advantage.Convergence figure of LGSCPSO comparing with original OPSO forf
1
–
f
8.
(a)
Number 1
(b)
Number 2
(c)
Number 3
(d)
Number 4
(e)
Number 5
(f)
Number 6
(g)
Number 7
(h)
Number 8


## 4.1. Test Functions

To illustrate the effectiveness and performance of LGSCPSO algorithm for
optimization problems, a set of 8 representative benchmark functions with
different dimensions were employed to evaluate it in comparison with original
PSO. Many authors tested algorithm using them widely.Sphere Model. Consider f
1
(
x
)
=
∑
i
=
1
n
x
i
2, where x
i
∈
[
-
100,100
].Schwefel’s Problem  1.2. Consider  f
2
(
x
)
=
∑
i
=
1
n
(
∑
j
=
1
i
x
j
)
2, where x
i
∈
[
-
100,100
].Schwefel’s Problem  2.21.  Consider f
3
(
x
)
=
max
⁡
i
{
|
x
i
|
,
1
≤
i
≤
n
}, where x
i
∈
[
-
100,100
].Generalized Rosenbrock’s Function. Consider f
4
(
x
)
=
∑
i
=
1
n
-
1
(
100
(
x
i
+
1
-
x
i
2
)
2
+
(
x
i
-
1
)
2
), where x
i
∈
[
-
100,100
].Ackley’s Function. Consider f
5
(
x
)
=
-
20
exp
⁡
(
-
0.2
(
1
/
n
)
∑
i
=
1
n
x
i
2
)
-
exp
⁡
(
(
1
/
n
)
∑
i
=
1
n
cos
⁡
2
π
x
i
)
+
20
+
e, where x
i
∈
[
-
32,32
].Generalized Griewank Function. Consider f
6
(
x
)
=
(
1
/
4000
)
∑
i
=
1
n
(
x
i
-
100
)
2
-
∏
i
=
1
n
cos
⁡
(
(
x
i
-
100
)
/
i
)
+
1, where x
i
∈
[
-
600,600
].Generalized Penalized Functions. Consider (17)
f
7
(
x
)
=
π
n
{
10
si
n
2
(
π
y
1
)
+
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
h
h
h
×
[
1
+
10
si
n
2
(
π
y
i
+
1
)
]
+
(
y
n
-
1
)
2
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
}
+
∑
i
=
1
n
u
(
x
i
,
10,100,4
)
,
f
8
(
x
)
=
0.1
{
si
n
2
(
π
y
1
)
+
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
h
h
h
h
×
[
1
+
10
si
n
2
(
π
y
i
+
1
)
]
+
(
y
n
-
1
)
2
∑
i
=
1
n
-
1
(
y
i
-
1
)
2
}
+
∑
i
=
1
n
u
(
x
i
,
10,100,4
)
, where (18)
u
(
x
i
,
a
,
k
,
m
)
=
{
k
(
x
i
-
a
)
m
,
x
i
>
a
0
,
-
a
≤
x
i
≤
a
k
(
-
x
i
-
a
)
m
,
x
i
<
-
a
,
y
i
=
1
+
(
1
4
)
(
x
i
+
1
)
,
x
i
∈
[
-
50,50
]
.They can be divided into unimodal (functionf
1
–
f
3) and multimodal functions (function f
4
–
f
8), where the number of local minima increases exponentially
with the problem dimension.


## 4.2. Experimental Results and Comparison

The experimental results of original PSO and LGSCPSO algorithm on each test
function are listed in Table1. To get
the average performance of the LGSCPSO algorithm ten runs on each problem
instance were performed and the solution quality was averaged. The comparisons
results of LGSCPSO and Original PSO algorithm are shown in Table 1 of the appendix.Table 1
The comparison results of the PSO algorithm and the LGSCPSO
algorithm.
Min./Max./Average (e
~
n represents ×
10
n)
a
w
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
1,
Dim 150,
Best 0;
PS 150,
EG 2500
.1.2.3.4.5.6.7
.8 .9
7.9247/798.1366/119.27351.0216/328.5215/36.36210.1788/3.6762/0.95800.0276/1.0302/0.17320.0015/0.2004/0.0345
6.199e  −  4/20.4397/2.1839 8.7194e  −  4/32.076/3.51180.0027/1.0443/0.37940.0301/3.1973e  +  5/5.7818e  +  4
4.8684/788.0086/137.77270.2372/39.35/5.29250.0516/72.579/18.80860.0024/16.9442/2.0285
0.0011/6.0067/0.7585 0.1109/326.9967/38.2332217.0564/2.5137e  +  3/922.9063131.8696/1.0093e  +  3/605.76083.5023/3.5197e  +  5/7.5222e  +  4
3.4298/466.135/129.40390.2165/36.7093/9.15460.0446/0.9215/0.32450.0081/5.1882/0.60590.0021/15.3744/3.103
9.091e  −  4/50.1812/9.9065 0.1802/416.0314/88.517622.395/2.7571e  +  3/760.493.3725/3.3151e  +  5/3.3313e  +  4
6.2134/108.3578/34.96780.336/297.5513/35.94330.0931/2.9157/0.69560.0255/12.8562/1.50620.0042/1.563/0.2760.0031/1.3039/0.21721.5877e  −  4/2.5115/0.3145
4.9801e  −  5/0.0052/8.8123e  −  4*9.7207e  −  5/390.2679/40.2488
202.8074/3.8168e  +  3/899.134725.8671/126.0089/74.35552.0381/42.8/14.73930.2738/12.3747/2.66930.0354/0.9538/0.21240.0088/0.3742/0.0659
0.0027/0.3203/0.05540.0104/0.1721/0.06780.5088/4.2984e  +  5/2.6068e  +  5
PSO
3.5545e  +  5/4.2006e  +  5/3.9996e  +  5
3.6563e  +  5/4.1904e  +  5/3.9654e  +  5
3.8295e  +  5/4.1465e  +  5/4.0243e  +  5
0.5082/4.185e  +  5/3.4273e  +  5
3.834e  +  5/4.2439e  +  5/4.0552e  +  5
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
2,
Dim 30,
Best 0;
PS 150,
EG 2000
.1.2.3
.4 .5.6.7.8.9
1.5314e  −  5/0.0012/2.3471e  −  47.544e  −  6/4.1057e  −  4/1.2064e  −  4
3.3468e  −  6/2.5205e  −  4/8.7442e  −  52.2108e  −  5/3.3094e  −  4/9.2148e  −  51.4607e  −  5/3.3855e  −  4/9.0081e  −  51.4749e  −  4/0.0097/0.00162.2105e  −  4/0.0705/0.02320.0879/0.9873/0.47321.3288/15.724/5.0923
3.4819e  −  8/9.7461e  −  7/2.8081e  −  73.6274e  −  9/2.252e  −  7/6.7068e  −  81.7849e  −  10/6.8322e  −  8/1.3459e  −  89.6468e  −  11/1.3930e  −  7/2.0447e  −  8
7.8029e  −  11/9.93e  −  9/4.5009e  −  93.0876e  −  9/4.2278e  −  7/5.6384e  −  81.8076e  −  7/1.0146e  −  5/2.1222e  −  61.437e  −  6/6.4389e  −  4/2.8104e  −  40.0042/0.1077/0.0241
1.6392e  −  9/8.0260e  −  8/2.4483e  −  84.3649e  −  11/1.9808e  −  8/3.0528e  −  94.3800e  −  11/8.3452e  −  9/2.5903e  −  9
2.9752e  −  12/5.6342e  −  10/1.6065e  −  10* 7.1437e  −  12/4.1282e  −  9/8.6320e  −  102.0796e  −  11/5.0013e  −  9/1.0295e  −  91.1123e  −  10/2.6655e  −  8/8.6717e  −  98.3474e  −  8/7.0419e  −  6/2.2156e  −  67.8655e  −  5/0.0055/0.002
1.9438e  −  8/4.5893e  −  6/1.0199e  −  66.9751e  −  9/4.4344e  −  6/5.5416e  −  73.0993e  −  10/2.9669e  −  8/6.9531e  −  9
5.1336e  −  11/6.7099e  −  8/9.4611e  −  91.2011e  −  10/6.3320e  −  9/1.7701e  −  94.0553e  −  10/3.5928e  −  8/5.5263e  −  96.2429e  −  9/6.8153e  −  7/1.8691e  −  79.0305e  −  7/1.0765e  −  4/1.9495e  −  52.6371e  −  4/0.0057/0.0024
0.0016/0.0746/0.01814.1506e  −  5/0.002/5.1297e  −  41.3450e  −  5/1.3095e  −  4/6.5852e  −  51.1121e  −  6/2.0099e  −  5/6.5348e  −  61.5491e  −  6/1.6945e  −  5/6.7154e  −  6
3.3047e  −  7/7.1073e  −  5/2.0406e  −  52.4160e  −  5/5.8853e  −  4/1.5538e  −  46.3392e  −  4/0.0096/0.00420.0595/1.9094/0.3682
PSO
364.253/2.4939e  +  4/1.0134e  +  4
1.3765/4.6947e  +  3/481.8464
0.0691/2.5876/0.7283
0.1723/3.9289/1.8005
17.3828/984.8965/188.1513
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
3,
Dim 30,
Best 0;
PS 150,
EG 2000
.1.2.3
.4 .5.6.7.8.9
2.3621e  −  4/0.0013/7.311e  −  46.0977e  −  5/4.2674e  −  4/2.1957e  −  45.5341e  −  5/7.0551e  −  4/2.1313e  −  4
4.9251e  −  5/3.4147e  −  4/1.4638e  −  45.2814e  −  5/0.0019/4.5195e  −  44.8841e  −  4/0.0053/0.00290.0046/0.0476/0.01910.0569/0.5819/0.22660.4589/4.2785/1.3995
1.3902e  −  5/2.0080e  −  4/8.8198e  −  51.1701e  −  6/2.8489e  −  5/8.0878e  −  63.8603e  −  7/8.6707e  −  6/1.7948e  −  6
1.4e  −  7/5.6791e  −  6/1.3055e  −  61.6044e  −  7/1.6259e  −  6/5.7405e  −  71.7168e  −  7/6.7027e  −  6/2.6332e  −  69.6091e  −  6/7.5094e  −  5/2.8845e  −  58.2486e  −  5/0.0012/3.9755e  −  40.001/0.0436/0.0158
1.6884e  −  5/3.3877e  −  4/1.194e  −  47.4212e  −  7/4.1010e  −  6/1.8869e  −  65.1647e  −  8/1.5398e  −  6/3.7869e  −  7
1.5353e  −  9/1.8722e  −  7/6.9333e  −  8* 4.7963e  −  9/3.1684e  −  7/6.3918e  −  82.0730e  −  9/6.1577e  −  7/9.0325e  −  83.4465e  −  8/7.9999e  −  6/1.0596e  −  61.1441e  −  6/1.3502e  −  5/5.7652e  −  62.4963e  −  5/0.0027/6.1308e  −  4
1.5547e  −  5/6.6884e  −  4/1.4984e  −  43.5838e  −  7/8.3143e  −  6/4.7977e  −  63.9851e  −  8/2.4747e  −  6/7.3211e  −  73.4271e  −  8/4.2062e  −  7/1.0447e  −  76.6274e  −  9/8.7745e  −  7/1.4709e  −  7
2.0358e  −  9/4.6960e  −  8/1.9028e  −  83.9048e  −  8/3.9274e  −  7/1.5350e  −  77.5274e  −  7/1.2135e  −  5/3.6172e  −  64.4217e  −  5/5.6209e  −  4/1.7621e  −  4
0.001/0.0262/0.00551.1206e  −  5/5.1122e  −  4/1.9353e  −  43.1555e  −  6/3.1428e  −  5/1.2056e  −  51.8118e  −  7/1.3756e  −  5/3.7208e  −  62.2951e  −  7/1.1701e  −  6/5.3581e  −  7
8.2804e  −  8/2.1413e  −  5/3.1923e  −  61.0194e  −  6/1.0954e  −  5/3.7688e  −  69.2958e  −  6/1.813e  −  4/1.0582e  −  40.0016/0.0311/0.0073
PSO
0.9442/7.2642/3.7178
0.1348/0.5193/0.2436
0.005/0.0675/0.0212
0.0084/0.043/0.03
0.2007/0.9216/0.4999
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
4,
Dim 25,
Best 0;
PS 150,
EG 2500
.1.2.3
.4 .5.6.7.8.9
1.8571/1.229e  +  3/135.88070.1568/2.5441e  +  3/388.20910.6066/799.8443/169.64220.2485/181.0596/25.4415
0.0237/40.8918/11.14410.0253/276.2887/57.01080.0916/228.4057/30.79650.4745/435.3273/56.96344.3957/341.7015/67.9295
6.2873/3.2863e  +  3/709.82220.0113/789.7377/115.4752.3034/796.4533/123.7322
9.6966e  −  5/6.8529/2.0791* 0.0069/23.873/4.99742.547e  −  4/9.0536/4.0420.0702/14.2596/6.66970.0189/16.2452/9.93793.2415/838.6487/128.4819
0.2013/285.8471/44.284.0693/4.9375e  +  3/562.5381
3.3983e  −  4/302.2135/32.97580.2062/179.2151/21.4179
3.5267e  −  4/3.8396/0.84943.6341e  −  4/6.6633/3.65670.0372/152.4195/22.09720.1941/238.9312/38.12250.7361/334.4369/49.1472
0.5944/144.3777/23.71881.8021/937.5994/153.76820.0455/207.5395/40.31320.0121/42.4684/5.2882
0.0018/182.335/21.08311.4713/10.0475/5.65290.1549/12.1124/7.78820.5691/1.2973e  +  3/170.2970.2681/312.4161/40.7414
0.0678/486.002/165.06634.3764/7.2066e  +  3/1.0582e  +  30.063/197.4164/32.77493.4459/962.925/115.1339
0.0125/423.4865/75.33540.1299/26.4581/10.28620.1671/672.3137/106.70180.6944/213.5931/30.54170.2055/1.1812e  +  3/161.6274
PSO
4.3221e  +  4/5.1189e  +  4/4.6655e  +  4
3.9971e  +  4/5.4303e  +  4/4.6593e  +  4
2.876e  +  4/5.5191e  +  4/4.7462e  +  4
3.937e  +  4/5.2589e  +  4/4.6619e  +  4
8.4709/102.7603/23.237
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
5,
Dim 30
Best 0;
PS 100,
EG 2000
.1.2
.3 .4.5.6.7.8.9
7.9936e  −  15/4.3521e  −  14/2.256e  −  147.9936e  −  15/7.9936e  −  15/7.9936e  −  154.4409e  −  15/7.9936e  −  15/7.2831e  −  15
4.4409e  −  15/7.9936e  −  15/5.1514e  −  154.4409e  −  15/7.9936e  −  15/5.8620e  −  154.4409e  −  15/2.3162/0.23164.4409e  −  15/1.1908e  −  9/1.1908e  −  106.9209e  −  5/2.2201/0.73284.8998e  −  5/1.6462/0.3991
7.9936e  −  15/7.9936e  −  15/7.9936e  −  154.4409e  −  15/7.9936e  −  15/5.1514e  −  15
4.4409e  −  15/7.9936e  −  15/5.5067e  −  15* 4.4409e  −  15/1.6462/0.28014.4409e  −  15/2.3162/0.59674.4409e  −  15/2.3162/1.09347.9936e  −  15/2.887/1.10890.0021/3.0268/1.75710.0043/2.5791/1.7115
4.4409e  −  15/7.9936e  −  15/7.2831e  −  15
4.4409e  −  15/7.9936e  −  15/5.1514e  −  154.4409e  −  15/1.5017/0.15024.4409e  −  15/2.0119/0.31674.4409e  −  15/1.6462/0.71284.4409e  −  15/3.6807/1.35854.4409e  −  15/2.4949/1.25647.9936e  −  15/2.0133/1.23727.9936e  −  15/2.6602/0.8856
7.9936e  −  15/1.5099e  −  14/9.4147e  −  154.4409e  −  15/7.9936e  −  15/7.2831e  −  15
4.4409e  −15/7.9936e  −  15/5.5067e  −  15* 4.4409e  −  15/2.5795/0.37354.4409e  −  15/2.0133/0.54.4409e  −  15/1.3404/0.22724.4409e  −  15/1.3404/0.1344.4409e  −  15/2.0133/0.54.4409e  −  15/1.1551/0.1155
1.056e  −  11/2.0851e  −  10/7.4168e  −  111.8652e  −  14/4.1656e  −  13/9.1838e  −  147.9936e  −  15/1.5099e  −  14/8.7041e  −  15
4.4409e  −  15/7.9936e  −  15/5.862e  −  15
4.4409e  −  15/7.9936e  −  15/5.862e  −  15 4.4409e  −  15/7.9936e  −  15/6.9278e  −  154.4409e  −  15/7.9936e  −  15/7.6383e  −  154.4409e  −  15/7.9936e  −  15/7.6383e  −  157.9936e  −  15/7.9936e  −  15/7.9936e  −  15
PSO
2.4655e  −  5/2.2315/1.1133
1.6387e  −  4/2.8859/0.7401
7.9936e  −  15/2.4959/0.6931
4.4409e  −  15/7.9936e  −  15/7.2831e  −  15
7.9936e  −  15/1.5099e  −  14/1.2967e  −  14
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
6,
Dim 150,
Best 0;
PS 150,
EG 2500
.1.2.3.4.5.6.7.8
.9
0.9416/1.4223/1.16350.2941/0.9446/0.48420.0383/0.2148/0.09940.002/0.3372/0.09454.6926e  −  4/0.2611/0.0958
3.7473e  −  5/0.1647/0.05411.572e  −  4/0.0664/0.01941.1609e  −  4/0.2322/0.05410.0129/3.4498e  +  3/639.0998
0.5782/1.452/0.94520.0804/2.7225/0.50280.0036/0.8091/0.15716.6171e  −  4/1.0183/0.1775
8.1220e  −  5/2.1079/0.41950.2266/6.0141/2.12031.6879/24.077/11.22931.9272/17.0006/7.12860.2582/3.7637e  +  3/668.9854
0.3887/8.8918/1.72370.0397/2.1197/0.46020.0055/4.4603/0.55018.199e  −  4/0.4379/0.122
2.5102e  −  4/0.5375/0.31580.0023/3.2548/0.76760.2953/4.2413/1.16681.3151/17.8887/9.72290.0979/12.3986/3.8033
0.9863/3.3689/1.32780.3404/0.6563/0.48770.0084/0.2726/0.13320.0013/0.2891/0.13430.0044/0.9635/0.18523.0448e  −  4/1.2816/0.26870.0074/0.5053/0.14691.3352e  −  5/0.061/0.0226
5.0879e  −  6/1.1096/0.1934*
3.0738/12.5482/6.15131.1998/2.6794/1.50130.545/1.2007/0.82910.0217/6.2987/0.80920.0061/0.2465/0.06210.0015/0.3055/0.0567
2.6726e  −  4/0.1051/0.02414.8173e  −  4/0.0609/0.0190.0283/3.9697e  +  3/2.3201e  +  3
3.1058e  +  3/3.9496e  +  3/3.6383e  +  3
2.766e  +  3/3.952e  +  3/3.5857e  +  3
0.9187/3.9622e  +  3/2.9803e  +  3
0.1916/3.9592e  +  3/2.8647e  +  3
3.5377e  +  3/3.9083e  +  3/3.7889e  +  3
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
7,
Dim 100,
Best 0;
PS 150,
EG 2500
.1.2.3.4.5.6
.7 .8
0.0014/0.1015/0.02684.7654e  −  5/0.0311/0.00343.3537e  −  5/0.0628/0.0165
5.8595e  −  7/0.0311/0.00368.2125e  −  7/0.1685/0.0298
5.8595e  −  7/0.0311/0.00360.0022/2.5577/0.824630.7659/1.0297e  +  9/3.6691e  +  8
2.1663e  −  5/0.1245/0.01896.125e  −  7/0.0622/0.00977.1955e  −  9/0.0311/0.0062
4.1191e  −  9/0.1555/0.02495.3424e  −  8/0.3743/0.07174.1946e  −  8/1.731/0.37940.1557/2.8177/0.89953.3049e  −  4/2.7669e  +  8/2.7669e  +  7
1.0775e  −  5/0.1866/0.03142.6987e  −  7/0.1557/0.02499.1232e  −  9/0.1868/0.03742.9509e  −  9/0.1868/0.04675.2224e  −  7/0.6882/0.2318
5.6598e  −  10/1.3103/0.28724.3761e  −  5/1.3392/0.41911.3321e  −  5/1.4361/0.5313
3.9207e  −  4/0.3284/0.08883.7171e  −  6/0.2498/0.06053.4490e  −  7/0.1244/0.04673.8808e  −  9/0.499/0.08724.4126e  −  9/0.2497/0.06553.7644e  −  9/0.2808/0.0779
1.0263e  −  10/0.2490/0.0498* 8.0965e  −  6/0.2789/0.0448
0.8393/18.623/5.1570.2215/1.3457/0.6130.0018/0.5005/0.16016.0798e  −  4/0.3451/0.09477.0408e  −  6/0.137/0.0287
3.7725e  −  7/0.4987/0.14056.0553e  −  7/0.5295/0.08610.4766/3.2293e  +  3/326.8969
2.2171e  +  9/2.6638e  +  9/2.3786e  +  9
2.05e  +  9/2.6165e  +  9/2.2883e  +  9
1.9676e  +  9/2.8563e  +  9/2.3233e  +  9
1.3125e  +  9/2.6573e  +  9/2.1146e  +  9
1.8631e  +  9/2.8019e  +  9/2.2317e  +  9
LGC
0.1
0.2
0.3
0.4
0.5
Fun
f
8,
Dim 100,
Best 0;
PS 150,
EG 2500
.1.2.3.4.5
.6 .7.8
0.0014/0.1008/0.01872.0521e  −  4/0.0132/0.00399.8083e  −  5/0.9506/0.1062
3.0775e  −  7/0.0037/5.5843e  −  43.3459e  −  6/4.3813e  −  4/1.47e  −  42.891e  −  6/2.5064/0.4977.7979e  −  4/9.103/2.6232.0949e  +  9/2.7924e  +  9/2.4913e  +  9
6.9482e  −  5/0.1026/0.01148.9263e  −  7/8.6653e  −  5/2.0482e  −  52.1133e  −  7/4.1364e  −  5/9.8935e  −  6
1.8204e  −  8/5.2730e  −  5/1.0100e  −  54.9008e  −  8/6.3249e  −  4/8.0700e  −  51.1649e  −  7/1.8877/0.50092.1485e  −  4/2.7725/0.87362.1328e  +  /2.8009e  +  9/2.5344e  +  9
1.9512e  −  5/0.0023/7.8123e  −  41.3466e  −  6/0.2978/0.03072.7567e  −  8/1.1156e  −  4/2.2848e  −  5
8.9806e  −  10/1.4026e  −  4/2.3733e  −  51.4968e  −  9/0.5878/0.06884.0322e  −  7/3.0954/0.86182.6362e  −  6/6.9698/2.03451.816e  +  9/2.8674e  +  9/2.4126e  +  9
2.037e  −  4/0.1195/0.01836.6231e  −  5/0.0019/5.873e  −  41.1378e  −  6/0.0015/2.6495e  −  41.8884e  −  7/0.0905/0.00915.5391e  −  10/8.5755e  −  5/9.8527e  −  6
4.9993e  −  10/7.1912e  −  5/8.2796e  −  6* 3.5393e  −  9/0.0011/1.6126e  −  41.6523e  −  7/2.1877/0.3866
5.8424/86.3769/26.71520.4244/9.4505/2.60560.0034/3.0968/0.5817.1911e  −  5/1.4867/0.1498
7.1442e  −  7/0.0052/0.00171.0077e  −  6/4.2974e  −  4/7.1607e  −  51.0331e  −  5/0.2203/0.04861.2518/2.391e  +  9/2.391e  +  8
PSO
1.8688e  +  9/2.8661e  +  9/2.3999e  +  9
2.05e  +  9/2.6165e  +  9/2.2883e  +  9
1.5174e  +  9/2.7626e  +  9/2.2677e  +  9
1.3125e  +  9/2.6573e  +  9/2.1146e  +  9
1.8631e  +  9/2.8019e  +  9/2.2317e  +  9
Remark: In Table1,
Fun and Dim denote function
and its dimension, respectively; the Best is this
function’s optimum value; PS and
EG indicate the population size and algorithm
terminate generation. PSO and LGC denote the original PSO and
LGSCPSO algorithm. The better solutions and the parameters found in
LGSCPSO are illustrated with bold letters, and the best solutions
are shown in data with asterisk.From Table1 one can observe that w
∈
[
0.2,0.4
] have the highest performance since using them has smaller
minimum and smaller arithmetic mean in relation to the solutions obtained by the
others, especially w
≈
0.4 has the better search efficiency. In LGSCPSO algorithm,
various parameter α should be selected for optimizing different
problems, as a whole, it is better when α
∈
[
0.3
0.9
], especially α
≈
0.4. From simulation results we can obtain that the LGSCPSO
algorithm is clearly better than the original PSO for continuous nonlinear
function optimization problem. The convergence figures of most effective LGSCPSO
comparing with original PSO for 8 instances are as follows.From Figure1 we can discover that the
convergence rate of LGSCPSO algorithm is clearly faster than the OPSO on every
benchmark function, especially it is more efficacious than OPSO for middle and
large size optimization problem. Accordingly, we can state that the LGSCPSO
algorithm is more effective than OPSO algorithm. For test function 5, the search
capability of LGSCPSO algorithm is worse than that of OPSO at the beginning of
the iteration. However, seen from the enlarged figure, the search capability of
LGSCPSO is markedly enhanced along with the iteration increasing. In addition we
can find that the simulation data from test function 5 has a strong
advantage.Convergence figure of LGSCPSO comparing with original OPSO forf
1
–
f
8.
(a)
Number 1
(b)
Number 2
(c)
Number 3
(d)
Number 4
(e)
Number 5
(f)
Number 6
(g)
Number 7
(h)
Number 8


## 5. Conclusions and Perspectives

According to the shortcoming of OPSO algorithm that it frequently gets the local
solution especially solving the middle or large size problem, a LGSCPSO algorithm is
presented, and simulations show that it is efficacious. The performance of the new
approach is evaluated in comparison to OPSO algorithm for eight representative
instances with different dimensions and obtained results show that LGSCPSO algorithm
is efficacious for solving optimization problems.The proposed LGSCPSO algorithm approach in this paper can be considered as effective
mechanisms from this point of view. There are a number of research directions that
can be considered as useful extensions of this research. Although the proposed
algorithm is tested with eight representative instances, a more comprehensive
computational study should be made to test the efficiency of proposed solution
technique. In the future it is maybe used for solving other discrete combinatorial
optimization problems such as traveling salesman problem and scheduling.



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*Source: 905712-2014-03-16.xml*