Received July 16, 2016, accepted August 8, 2016, date of publication September 1, 2016, date of current version January 27, 2017.
Digital Object Identifier 10.1109/ACCESS.2016.2604738
Convergence Analysis and Improvement of the
Chicken Swarm Optimization Algorithm
DINGHUI WU, SHIPENG XU, AND FEI KONG
Key Laboratory of Advanced Process Control for Light Industry, Ministry of Education, School of Internet of Things Engineering,
Jiangnan University, Wuxi 214122, China
Corresponding author: D. Wu (wdh123@jiangnan.edu.cn)
This work was supported by the National Natural Science Foundation of China under Grant 61572237 and Grant 61573167.
ABSTRACT In this paper, the convergence analysis and the improvement of the chicken swarm
optimization (CSO) algorithm are investigated. The stochastic process theory is employed to establish the
Markov chain model for CSO whose state sequence is proved to be finite homogeneous Markov chain and
some properties of the Markov chain are analyzed. According to the convergence criteria of the random
search algorithms, the CSO algorithm is demonstrated to meet two convergence criteria, which ensures the
global convergence. For the problem that the CSO algorithm is easy to fall into local optimum in solving
high-dimensional problems, an improved CSO is proposed, in which the relevant parameters analysis and
the verification of optimization capability are made by lots of test functions in high-dimensional case.
INDEX TERMS Chicken swarm optimization, Markov chain, state transition, global convergence,
benchmark function.
I. INTRODUCTION
The swarm intelligence optimization algorithm is a metaheuristic algorithm based on population, which finds the
optimal solution of the problem through cooperation and
competition between individuals within populations [1], [2].
The swarm intelligence optimization algorithm as a new
evolutionary computation technology has become the focus
of a growing number of scholars, and lots of researchers
have proposed many swarm intelligence algorithms for optimization problems, such as cat swarm optimization [3], the
firefly algorithm [4], the wolf search algorithm [5], the monkey search algorithm [6] and the bat algorithm [7]. These
algorithms can be used to solve most optimization problems. The application field of these algorithms has been
extended to neural network training [8]–[10], multi-objective
optimization [11]–[13], data clustering [14], pattern recognition [15], [16], robot control [17], [18], dada mining [19], [20]
and system identification [21]–[23] etc. However, the basis of
these algorithms comes from the simulation of the biological
community, which lacks relevant theoretical analysis. What
is more, the parameter settings are empirically determined
without precise theoretical basis. So it is important to make
the theoretical research on the swarm intelligent optimization
algorithms.
The chicken swarm optimization is a meta-heuristic algorithm based on population, which mimics the behaviors of
the chicken swarm [24]. The whole swarm can be divided
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into several groups, each of which consists of one rooster
and many hens and chicks. Different chickens follow different
laws of motions. In particular hierarchy order, there is a competition between the various subgroups. But so far, no literature was found about the relevant theoretical research (such
as convergence analysis) on the chicken swarm optimization
algorithm. According to the problem that the chicken swarm
optimization is easy prone to premature convergence in solving high-dimensional optimization problems, an improved
chicken swarm optimization is proposed and the optimization
ability is tested by eight benchmark functions [25]. But the
relevant parameter analysis was not made, and test functions
used were too little, which cannot fully validate the optimization ability of the proposed algorithm. So there is an urgent
need for analyzing the convergence of the chicken swarm
optimization. In addition, based on the work in [25], the relevant parameters analysis and the verification of optimization
capability by lots of test functions are also necessary.
The Markov chain is a random process that plays an important role and has universal significance. It has a wide range of
applications. The Markov chain theory has a strong capability
in terms of convergence analysis of randomized algorithms
and probabilistic analysis, and has been successfully applied
to the simulated annealing [26], the PSO [27], the ant colony
algorithm [28] and the artificial bee colony algorithm [29].
In this paper, the Markov chain is applied to convergence
analysis of the chicken swarm optimization, by establishing
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VOLUME 4, 2016
D. Wu et al.: Convergence Analysis and Improvement of the CSO Algorithm
the Markov chain model of the chicken swarm optimization to
study the state metastatic behavior of the chicken swarm optimization. Convergence performance of the chicken swarm
optimization was also analyzed according to randomized
algorithms convergence criteria. Based on the work in [25],
the relevant parameter analysis and the verification of the
optimization capability are also made by many test functions
in high-dimensional case.
The rest of this paper is organized as follows. In Section 2,
the chicken swarm optimization is introduced. The details
about the Markov chain model of the chicken swarm optimization are discussed in Section 3. The convergence analysis
of the chicken swarm optimization is presented in Section 4.
Section 5 gives the details of the improved chicken swarm
optimization and the improved chicken swarm optimization
for test functions is described in Section 6. In Section 7, some
conclusions and discussions are given.
II. THE CHICKEN SWARM OPTIMIZATION
The chicken swarm optimization mimics the hierarchal order
in the chicken swarm and the behaviors of the chicken swarm,
which stems from the observation of the birds foraging behavior. The chicken swarm can be divided into several groups,
each of which consists of one rooster, many hens, and several
chicks. When in foraging, the roosters can always find food
preferentially. The hens always follow the roosters to look for
food, and the chicks followed their mother in search of food.
The different individuals within the chicken swarm follow
different laws of motions. There exist competitions between
the different individuals within the chicken swarm under the
particular hierarchal order.
The position of each individual within the chicken swarm
represents a feasible solution of the optimization problem.
First define the following variables before describing location
update formula of the individuals within the chicken swarm:
RN , HN , CN and MN are the number of roosters, hens, chicks
and mother hens, respectively; N is the number of the whole
chicken swarm, D is the dimension of the search space;
xi,j (t) (i ∈ [1, · · · , N ], j ∈ [1, · · · , D]) is the position of each
individuality time t.
In the chicken swarm, the best RN chickens would be
assumed to be the roosters, while the worst CN ones would
be regarded as the chicks. The rest of the chicken swarm is
viewed as the hens. Roosters with best fitting values have
priority for food access than the ones with worse fitting
values. The position update equations of the roosters can be
formulated below:
xi,j (t + 1) = xi,j (t)(1 + Randn(0, σ 2 )),
(1)

1,
fi ≥ fk , k ∈ [1, N ], k 6 = i,
fk − fi
σ2 =
exp(
), otherwise,
|fi | + ε
(2)
where Randn(0, σ 2 ) is a Gaussian distribution with mean 0
and standard deviation σ , ε is the small constant to avoid
VOLUME 4, 2016
zero-division-error, and k is a rooster’s index, which is
randomly selected from the roosters group (k 6 = i), fi is the
fitness value of particle i.
As for the hens, they can follow their group-mate roosters
to search for food and randomly steal the food found by other
the individuals. The position update equations of the hens are
as follows:
t+1
t
t
t
Xi,j
= Xi,j
+ C1 Rand(Xr1,j
− Xi,j
)
t
t
− Xi,j
),
+ C2 Rand(Xr2,j
(3)
C1 = exp((fi − fr1 )/(|fi | + ε)),
(4)
C2 = exp((fr2 − fi )),
(5)
where Rand is a uniform random number over [0, 1], r1 is an
index of the rooster, which is the ith hen’s group-mate, and
r2 is the index of the chicken (rooster or hen), which is
randomly chosen from the chicken swarm r1 6 = r2 .
With respect to the chicks, they follow their mother to
forage for food. The position update equation of the chicks
is formulated below:
t+1
t
t
t
Xi,j
= Xi,j
+ F(Xm,j
− Xi,j
).
(6)
where xm,j (t) is the position of the chick’s mother,
F ∈ [0, 2] is the follow coefficient, which indicates that the
chick follows its mother to forage for food.
III. THE MARKOV CHAIN MODEL OF THE
CHICKEN SWARM OPTIMIZATION
For convenience, the dimension of the particle is not considered here. The simplified position update formulas are as
follows. The position update formula of the roosters is
xi (t + 1) = xi (t)(1 + R1 ).
(7)
where R1 is the fixed value between 0 and 1. The position
update formula of the hens is
xi (t + 1) = xi (t) + C1 R2 (xr1 (t) − xi (t))
+ C2 R3 (xr2 (t) − xi (t)),
(8)
where R2 and R3 are the fixed values between 0 and 1,
C2 < 1 < C1 .
The position update formula of the chicks is
xi (t + 1) = xi (t) + F(xm (t) − xi (t)).
(9)
Several definitions and mathematical description to illustrate
the Markov chain model of the chicken swarm optimization
are given in the following [31].
Definition 1 (The State of the Chicken and the State Space
of the Chicken): The chicken state consists of the location
of the chicken in the chicken swarm, denoting x. x ∈ A,
where A is the feasible solution space. All the possible states
of the chickens constitute the state space of chicken, denoting
X = {x|x ∈ A}.
Definition 2 (The State of the Chicken Swarm and the State
Space of the Chicken Swarm): The states of all the chickens
in the chicken flock constitute the state of the chicken swarm,
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D. Wu et al.: Convergence Analysis and Improvement of the CSO Algorithm
denoting s = (x1 , x2 , · · · , xN ), where xi is the state of the
ith chicken and N is the total number of the individuals in the
chicken swarm. All the states of the chicken swarm constitute
the state space of the chicken swarm, and it is remembered as
S = {s = (x1 , x2 , · · · , xN )|xi ∈ A, 1 ≤ i ≤ N }.
Definition P
3 (The State Equivalence): For s ∈ S and x ∈ s,
let ϕ(s, x) = N χ|x|(xi ) , where χA is the characteristic function of event A and ϕ(s, x) are the individuals in the chicken
swarm state s contained in the state x. For two chicken swarm
states s1 , s2 ∈ S, x ∈ A, if ϕ(s1 , x) = ϕ(s2 , x), s1 is called to
be equivalent s2 , denoting s1 ∼ s2 .
Definition 4 (The State Equivalence Class): The chicken
swarm state equivalence class induced by equivalence relation in S calls L = S/ ∼ or chicken swarm equivalence class
for short.
The chickens equivalence class has the following
properties.
1) Any chicken swarm states in the equivalence class L
are equivalent each other.
2) Any one in is not equivalent with the one outside L.
3) There is no intersection between one equivalence class
and another.
Definition 5 (The State Transition of the Chicken): In the
chicken swarm optimization, for xi ∈ s, xj ∈ s, the state of
the chicken xi one step transmits to xj , denoting Ts (xi ) = xj .
Theorem 1: In the chicken swarm optimization, transition
probability of the state of the chicken xi one step transition
to xj , namely
p(Ts (xi ) = xj )


pr (Ts (xi ) = xj )
= ph (Ts (xi ) = xj )


pc (Ts (xi ) = xj )
achieved by the roosters,
(10)
achieved by the hens,
achieved by the chicks.
Proof: The chicken swarm is a set of points in hyperspace, so the chicken location update process is a transition between a set of points in hyperspace. According to
Definition 5 and the geometry nature of the chicken swarm
optimization, we can get one-step transition probability of the
roosters from xi to xj , namely

 1 , x ∈ [x , x + x R ],
j
i i
i 1
(11)
pr (Ts (xi ) = xj ) = |xi R1 |

0,
otherwise.
One-step transition probability of the hens from xi to xj is
ph (Ts (xi ) = xj ) =
1
,
|C1 R2 (xr1 − xi ) + C2 R3 (xr2 − xi )|
(12)
for xj ∈ [xi , xi + C1 R2 (xr1 − xi ) + C2 R3 (xr2 − xi )], 0 for
otherwise. One-step transition probability of the chicks from
xi to xj is
pc (Ts (xi ) = xj )

1

, xj ∈ [xi , xi + F(xm − xi )],
= |F(xm − xi )|
(13)

0,
otherwise.
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Definition 6 (The State Transition Probability of Chicken
Swarm Optimization): In the chicken swarm optimization,
for si ∈ S, sj ∈ S, the state of the chicken swarm si one
step transmits to sj , denoting TS (si ) = sj . The transition
probability of the state of the chicken swarm si one step
transition to sj is
p(Ts (si ) = sj ) =
N
Y
p(Ts (xin ) = xjn ),
(14)
n=1
where N is the number of the individuals in the chicken
swarm. That is to say, the transition probability of the state
of the chicken swarm si one step transition to sj is the sum
of the state transition probability of each individual of Si one
step transition to each individual of Sj .
Definition 7 (The Transition Probability of the State
Equivalence Class of Chicken Swarm): Assume that
Li = (si1 , si2 , · · · , sih ) and Lj = (si1 , si2 , · · · , sik ) are
the chicken swarm state equivalence class induced by the
equivalence relation in S. Li one-step transmits to Lj , denoting
TL (Li ) = Lj , one-step transition probability of which is
p(TL (Li ) = Lj ) =
k
h X
X
p(TS (Si f ) = Sj e).
(15)
f =1 e=1
It shows that the transition probability of the chicken swarm
equivalence class is the sum of the state transition probability
of each chicken swarm of Li one step transition to Lj .
IV. THE CONVERGENCE ANALYSIS OF THE
CHICKEN SWARM OPTIMIZATION
Definition 8 (The Markov Chain): Suppose that there is
a random process xn , n ∈ T , and the parameter set T is a
discrete time sequence, namely T = 0, 1, 2, · · ·. All possible
values x substitute the discrete state space I = {i0 , i1 , i2 , · · · }.
If the conditional probability p(xn+1 = in+1 |x0 = i0 ,
x1 = i1 , · · · , xn = in ) = p(xn+1 = in+1 |xn = in ) is met
for an arbitrary integer n ∈ T and {i0 , i1 , i2 , · · · , in+1 } ∈ I ,
{xn , n ∈ T } is called a Markov chain.
Definition 9 (The Finite Markov Chain): If the state space I
is finite, we call the chain the finite Markov chain.
Definition 10 (The Homogeneous Markov Chain):
p(xn+1 = in+1 |xn = in ) is the conditional probability that
the system state in in time n shifts to the new state in+1 . If the
probability is independent with time, we call the chain the
homogeneous Markov chain.
Theorem 2: The population sequence generated by the
chicken swarm optimization is a finite homogeneous Markov
chain.
Proof: First, prove the chicken swarm sequence is a
Markov process and then prove that the Markov chain satisfies the homogeneous and finiteness. According to Definition 6, among the chicken swarm state sequences s(t), t ≥ 0,
s(t) ∈ S and s(t + 1) ∈ S, the transition probabilities
p(TS (s(t)) = s(t + 1)) depends on the transition probabilities p(TS (x(t)) = x(t + 1)) of all the chickens in the
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D. Wu et al.: Convergence Analysis and Improvement of the CSO Algorithm
chicken swarm. From Theorem 1, the state transition probability p(TS (x(t)) = x(t + 1)) of each individual of the
chicken swarm is only related to the state x(t) of time t,
the random numbers R1 , R2 and R3 belong to [0, 1], the
random number F belongs to [0, 2], C1 , C1 , xr1 , xr2 and xm ,
so p(TS (x(t)) = x(t + 1)) is only associated with the
state at time t, not related to time t. In addition, according
to Definition 8, the population sequence generated by the
chicken swarm optimization possesses Markov sex. The sincere chicken swarm state space S is limited, according to
Definition 9, the population sequence {s(t), t ≥ 0} generated
by the chicken swarm optimization is a finite Markov chain.
From Theorem 1, p(TS (x(t)) = x(t + 1)) is only related to the
state at time t, and not associated with time t, so the population sequence produced by the chicken swarm optimization
is limited a homogeneous Markov chain. QED.
A. THE CONVERGENCE RULE
The chicken swarm optimization is a random search algorithm, so the convergence behaviors of the chicken swarm
optimization can be judged by the convergence rule of the
random search algorithm in [32]. For the optimization problem hA, f i, there exists a stochastic optimization algorithm D,
the iteration result of kth is xk , and the next iteration value is
xk+1 = D(xk , ζ ), where A is feasible solution space, f is the
fitting function and ζ is the iteration solution searched already
for the chicken swarm optimization.
Define the search infimum
α = inf{t|v(x ∈ A|f (x) < t) > 0},
(16)
where v(x) is the Lebesgue measure of the set x. Define the
optimal solution region
(
x ∈ A|f (x) < α + ε, α < ∞,
Rε,M =
(17)
x ∈ A|f (x) < −C,
α = −∞.
where ε > 0 and C is a sufficiently large positive. If the random algorithm D finds a point located in Rε,M , the algorithm
is thought to be approximate global optimal or acceptable
global optimal.
Condition H1 If ζ ∈ A, then f (D(x, ζ )) ≤ f (x).
Condition H2 For B ∈ A with v(B) > 0, there is
Q∞
k=0 (1−uk (B)) = 0, where uk (B) is the probability measure
of the kth iteration search solution of the algorithm D situated
in the set B.
Theorem 3 (The global convergence of the random
algorithm): Suppose that f is measurable and A is a measurable subset of Rn , if Conditions H1 and H2 are satisfied
and {xk }∞
k=0 is a sequence generated by the algorithm D, the probability measure P(xk ∈ Rε,M ) satisfies
limk→∞ P(xk ∈ ε, M ), where Rε,M is the optimal area. Then
the algorithm D is of the global convergence.
B. THE CONVERGENCE OF THE CHICKEN
SWARM OPTIMIZATION
Theorem 4: The chicken swarm optimization algorithm
satisfies Condition H1.
VOLUME 4, 2016
Proof: The chicken swarm optimization must update the
current best position of the individuals for iteration, namely
(
xi (t − 1), f (xi (t)) > f (xi (t − 1)),
xi (t) =
(18)
xi (t),
f (xi (t)) ≤ f (xi (t − 1)).
So that the chicken swarm optimization saves the best
position of the group for iteration and Condition H1 is
satisfied. QED.
Definition 11 (The Optimal State Set of Chicken Swarm):
Suppose that the optimal of the optimization problem hA, f i
is g∗ , define the optimal state set of chicken swarm as
G = {s = (x)|f (x) = f (g∗ ), s ∈ S}.
If G equals S, each solution in the feasible solution space
is not only a feasible solution but also an optimal one. At this
time, optimization is meaningless, so the following discussion is based on G ⊂ S.
Definition 12 (The Absorption State Markov Chain): Based
on the Markov chain {s(t), t ≥ 0} of the population sequence
of the chicken swarm optimization and the optimal state set
G ⊂ S, if {s(t), t ≥ 0} satisfies the conditional probability
p{xk+1 ∈
/ G|xk ∈ G} = 0, the Markov chain is called the
absorbing state Markov chains.
Theorem 5: For the chicken swarm optimization, the optimal state set G is a closed set of the state space S.
Proof: In the chicken swarm optimization, the best individual retention mechanism is adopted for the update strategy
of the optimal individual. That is to say, when the fitting
value of the current best individual is better than that of
the previous best individual, the previous individual will be
replaced, as shown in (18). Thus, this ensures that the new
individual is not worse than the old one for iteration of the
chicken swarm optimization, so the conditional probability
p(xk+1 ∈
/ G|xk ∈ G) = 0 is satisfied. That is, the population
sequence {s(t), t ≥ 0} generated by the algorithm is the
absorption state Markov chains. QED.
Theorem 6: For the chicken swarm optimization, the optimal chicken swarm state set G is a closed set of the state
space S.
Proof: For si ∈ G, si ∈
/ G, sj ∈ S, the state transiQN
tion probability TS (si ) = sj is p(TS (si ) = sj ) =
k=1 p
(TS (xik ) = xjk ). Thus at least one particle state is optimal.
Suppose that g∗ ∼ xi0 k is the optimal state. That is to say,
there exists at least xi0 k ∈ G such that p(TS (xi0 k ) = xjk ) = 0.
At this time p(TS (xi0 k ) = xjk ) = 0, so the optimal chicken
swarm state set G is a closed set of the state space S. QED
Theorem 7:
T There do not exist a non-empty closed set M
such that M G = ∅ in the chicken swarm state space S.
Proof: Suppose
that there is a non-empty closed set
T
subject to M G = ∅. Suppose that xi = (g∗ , g∗ , · · · ,
g∗ ) ∈ G and xj = (xj1 , xj2 , · · · , xjN ) such that
f (xjc ) > f (g∗ ). According to the Chapman-Kolmogorov
equation, it is concluded that
X
X
plSj ,Si =
···
p(TS (Sj ) = Sr1 )
Sr1 ∈S
Srl−1 ∈S
× p(TS (Sr1 ) = Sr2 ) · · · p(TS (Srl−1 ) = Si ).
(19)
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D. Wu et al.: Convergence Analysis and Improvement of the CSO Algorithm
By limited iterations for the chicken swarm optimization,
Equations (11)–(13) in Theorem 1 are satisfied. So when
the step size is large enough, each item one-step transition
probability p(Ts (xrc+j )) = xrc+j+1 of the expansion in (19)
satisfies Equation (10), namely p(Ts (xrc+j )) = xrc+j+1 > 0, so
does plSj ,Si . The conclusion can be achieved that is not a closed
set and contradicts the assumption. Therefore, the Markov
chain of the chicken swarm state is irreducible, and the state
space does not contain the non-empty closed set. QED.
Theorem 8 [33]: For Markov chain, suppose that there
is non-empty closed set E, but thereTdoes not exist another
non-empty closed set O makes E O = ∅. If j ∈ E,
then limk→∞ p(xk = j) = 5j , and if j ∈
/ E, then
limk→∞ p(xk = j) = 0.
Theorem 9: When the internal iteration of the chicken
swarm approaches to infinity, the chicken swarm state
sequence will enter the optimal state set.
Proof: By using Theorems 6 to 8, we can draw the
conclusion that Theorem 9 is true. QED.
Theorem 10: The chicken swarm optimization converges
to a global optimal solution.
Proof: Based on Theorem 4, the chicken swarm optimization satisfies Condition H1. Moreover, from Theorem 9,
the probability that the chicken swarm optimization does not
search a global
Q optimal solution after countless times is zero,
so there is ∞
k=0 (1 − uk [B]) = 0, where uk [B] is the probability measure of the kth iteration search solution of the algorithm situated in the set B. The chicken swarm optimization
satisfies Condition H2. In the chicken swarm optimization,
the best individual retention mechanism is adopted for the
update strategy of the optimal individual. That is to say, when
the fitting value of the current best individual is better than
that of the previous best individual, the previous individual
will be replaced with (18), otherwise, the original individual
is retained. That is, when the iteration goes to infinity, we
have limk→∞ P(xk ∈ Rε,M ) = 1, where {xk }∞
k=0 is an
iterative sequence produced by the chicken swarm optimization. According to Theorem 3, we can draw the conclusion
that the chicken swarm optimization is a global convergence
algorithm. QED.
V. THE IMPROVED CHICKEN SWARM OPTIMIZATION
A. THE PERFORMANCE ANALYSIS FOR THE
CHICKEN SWARM OPTIMIZATION
In [24], the standard functions are only tested in lowdimensional cases, without involving high-dimensional
cases. Here, the chicken swarm optimization (CSO) algorithm is adopted to test four high-dimensional benchmark
functions and the comparison with other two algorithms is
made. The results are shown in Table 1, where the boldfaces
indicate a minimal value obtained for three algorithms.
From Table 1, we can see that although the results
obtained by the chicken swarm optimization are better than
that by PSO and BA, it has significantly deviated from
the global optimum, especially for Rosenbrock, the algorithm obviously falls into a local optimum. From the above
discussion, we can draw the conclusion that the chicken
swarm optimization is easy to prone to premature convergence when in solving high-dimensional optimization
problems.
B. THE IMPROVED CHICKEN SWARM OPTIMIZATION
In order to overcome the premature convergence of the
chicken swarm optimization in solving high-dimensional
optimization problems, the location update formula of the
individuals within the chicken swarm should be improved.
For equation (6), the chicks only follow their mothers to
search for food, but do not follow the rooster of the subgroup. Therefore, the chicks can only obtain the position
information of their own mother. The chicks will fall into a
local optimum when their mothers fall into a local optimum.
So the position update equation for chicks must be modified.
The modified position update equation of the chicks is as
follows:
xi,j (t + 1) = wxi,j (t) + F(xm,j (t) − xi,j (t))
+ C(xr,j (t) − xi,j (t)).
(20)
TABLE 1. The experiment results for three algorithms at D = 100.
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D. Wu et al.: Convergence Analysis and Improvement of the CSO Algorithm
where m is the index of the mother of chick i, r is the index
of the rooster in the subgroup, C is the learning factor, which
indicates that the chicks learn from the rooster in the subgroup
and w is a self-learning coefficient for the chicks, which is
very similar to the inertia weight in PSO.
1) THE DETAILS FOR THE IMPROVED CHICKEN
SWARM OPTIMIZATION
The detail process for improved chicken swarm optimization
is formulated below:
Step 1 Initialize the chicken swarm x and the related
parameter (RN , HN , CN , MN , · · · ).
Step 2 Evaluate the fitting values of the chicken swarm x,
and initialize the personal best position pbest and the global
best position gbest.t = 1.
Step 3 If t mod G is 1, sort the fitting values of the individuals within the chicken swarm, and build the hierarchal
order of the chicken swarm; divide the whole chicken swarm
into several subgroups and ensure the relationship between
the hens and the chicks.
Step 4 Renew the position of the roosters, the hens and
the chicks using equations (1), (3) and (7), respectively, and
calculate the fitting values of the individuals.
Step 5 Update the personal best position pbest and the
global best position gbest.
Step 6 t = t + 1, if the iteration stop condition is
met, halt iteration and export the global optimum; otherwise,
go to step 3.
2) THE CONVERGENCE ANALYSIS OF THE IMPROVED
CHICKEN SWARM OPTIMIZATION
About the earlier work, Trelea analyzed the convergence
and parameter selection of the particle swarm optimization
algorithm [34]. Compared with the original chicken swarm
optimization, the position update equation of the chicks for
the improved chicken swarm optimization is modified. The
structure of the chicken swarm and domination relation of the
individuals has not changed, so the Markov chain model of
the improved chicken swarm optimization is the same as the
original chicken swarm optimization. The original chicken
swarm optimization has been proved to converge to the global
optimum by Theorem 10.
3) THE PARAMETER ANALYSIS
Firstly, four inertia weight (denote as w) update strategies are
described in [35], namely linear decreasing, opening upward
parabola descending, opening downward parabola decreasing
and exponentially descending. The update equations of the
inertia weight use equations (21)–(24). In the same iterations,
the algorithm produces different optimization results adopting different inertia weight update policies. Therefore, the
classic test functions namely Sphere, Rosenbrock, Griewank
and Rastrigrin are selected to verify the optimization performance of the algorithm when adopting different inertia
weight strategies. Parameter settings are as follows: D = 100,
Ngen = 500, Npop = 50, G = 10, Nrun = 50, wmax = 0.9,
wmin = 0.4, C = 0.4, RN = 0.2, HN = 0.6, MN = 0.1,
CN = N − RN − HN . The experiment results are shown
in Table 2.
As can be seen from Table 2, for Sphere, Rosenbrock
and Griewank, the improved chicken swarm optimization can
obtain a smaller mean and standard deviation and the average
run time is shortened by using exponentially descending
inertia weight strategy. For the test function Rastrigrin, the
improved chicken swarm optimization can get a smaller mean
and standard deviation by adopting exponential decreasing
inertia weight strategy, but the average run time of the algorithm is slightly more than the case of using other strategies.
The proposed algorithm can improve the convergence precision without affecting the convergence speed by employing
TABLE 2. The parameter estimates and errors (σ 2 = 0.102 , L = 2000).
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the exponential decreasing inertia weight strategy, so the
exponentially decreasing strategy is adopted to update the
inertia weight. The inertia weight w is exponentially decreasing from 0.9 to 0.4 with the increase of iterations. Four update
equations of w are as follows:
w = wmax − (wmax − wmin )gen/Ngen,
w = (wmax − wmin )(t/tmax )
(21)
2
+ (wmin − wmax )(2t/tmax ) + wmax ,
(22)
w = −(wmax − wmin )(gen/Ngen) + wmax ,
(23)
w = wmin (wmax /wmin )(1/(1+10t/Ngen)) ,
(24)
2
where wmin is the final value of learning coefficients, wmax is
the initial value of learning coefficients, t is the current value
of iterations, Ngen is the maximum of iterations.
Secondly, the parameter G should be taken to be an appropriate value for different optimization problems. If G is too
large, the algorithm cannot quickly converge to the global
optimum. On the contrary, if G is too small, the algorithm may
fall into a local optimum. In [24], G is given in a reasonable
range, but it did not give the details for the experiments. To do
this, the classic test functions Sphere, Rosenbrock, Griewank,
Rastrigrin and Ackley are selected to study the influence of
the different values of G on the optimization performance of
the algorithm. Since the values of G have a very wide range,
it is impossible to select all possible values of G to verify
the optimization performance of the algorithm. Here, G = 2,
G = 6, G = 10, G = 13, G = 16 and G = 20 are chosen to
validate the optimization performance of the algorithm.
The parameter settings for the improved chicken swarm
optimization are as follows: D = 100, Ngen = 200,
Npop = 50, Nrun = 50, wmax = 0.9, wmin = 0.4, C = 0.4,
RN = 0.2, HN = 0.6, MN = 0.1, CN = N − RN − HN . The
exponential decreasing strategy is adopted for inertia weight
of the improved chicken swarm optimization. The average
convergence curves of the five test functions are shown in
Figures 1 to 5, respectively, in which the abscissa represents
the maximum number of iterations and the ordinate denotes
the common logarithm of the fitting value.
As can be seen from Figure 1, when G = 10, the algorithm has faster convergence rate and higher convergence
FIGURE 1. The average convergence curve of Sphere function versus G.
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FIGURE 2. The average convergence curve of Rosenbrock function
versus G.
FIGURE 3. The average convergence curve of Griewank function versus G.
FIGURE 4. The average convergence curve of Rastrigrin function versus G.
accuracy. In Figure 2, synthesizing the convergence rate and
accuracy of the algorithm, when G = 10, the algorithm has
a higher convergence accuracy in ensuring the convergence
rate. In Figure 3, when G = 10, the algorithm has a faster
convergence speed and higher convergence accuracy. From
Figure 4, we can see that when G = 13, the algorithm
has a higher convergence accuracy. In Figure 5, in order
to obtain higher convergence accuracy, G should take 10.
In conclusion, taking into account all the standard test functions, and considering both the convergence rate and convergence accuracy of the algorithm, when G = 10, the improved
chicken swarm optimization shows better optimization capabilities for most test functions.
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D. Wu et al.: Convergence Analysis and Improvement of the CSO Algorithm
FIGURE 5. The average convergence curve of Ackley function versus G.
Furthermore, the learning factor (denote as C) has a significant impact on the optimization performance of the algorithm. If C is too large, the algorithm cannot quickly converge
to the global optimum. On the contrary, if C is too small,
the convergence accuracy of the algorithm will decrease.
Therefore, the classic test functions Sphere, Rosenbrock,
Griewank and Ackley are selected to study the influence of
the different values of C on the optimization performance of
the algorithm. In the paper, C = 0.2, C = 0.4, C = 0.6,
C = 0.8 and C = 1.0 were selected to validate the optimization performance of the algorithm.
The parameter settings for the improved chicken swarm
optimization are as follows: D = 10, Ngen = 200,
Npop = 50, Nrun = 50, wmax = 0.9, wmin = 0.4, G = 10,
RN = 0.2N , HN = 0.6N , MN = 0.1HN , CN = N −RN −HN .
An exponential decreasing strategy is adopted for the inertia
weight of the improved chicken swarm optimization. The
average convergence curves of the four test functions are
shown in Figures 6 to 9, where the abscissa represents the
maximum number of iterations and the ordinate denotes the
common logarithm of the fitting values.
FIGURE 7. The average convergence curve of Rosenbrock function
versus C.
FIGURE 8. The average convergence curve of Griewank function versus C.
FIGURE 9. The average convergence curve of Ackley function versus C.
FIGURE 6. The average convergence curve of Sphere function versus C.
As can be seen from Figure 6, when C = 0.4, the algorithm
has a higher convergence rate. In Figure 7, synthesizing convergence rate and accuracy of the algorithm, when C = 0.4,
the algorithm has a higher convergence accuracy. In Figure 8,
when C = 0.4, the algorithm has a faster convergence speed
and higher convergence accuracy. From Figure 9, we can see
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that when C = 0.4, the algorithm has a higher convergence
accuracy. On the whole, taking into account all the standard
test functions, and considering both the convergence rate and
convergence accuracy of the algorithm, when C = 0.4, the
improved chicken swarm optimization show better optimization capabilities for the all test functions.
VI. THE IMPROVED CHICKEN SWARM OPTIMIZATION
FOR TEST FUNCTIONS
A. THE PARAMETER SETTING
To test the performance of the improved chicken swarm optimization (CSO) algorithm, fourteen canonical benchmark
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TABLE 3. Fourteen benchmark functions.
FIGURE 10. The average convergence curve for F1.
test functions shown in Table 3 are used here for comparison [36]. In this table, all of test functions are minimum problems, n is the dimension of test functions, the first function is
a simple modal function (it has a single local optimum that is
a global optimum) while the other functions are multimodal
functions with many local optima. The amount of local optimum or saddles increases with increasing complexity of the
function s, i.e. with increasing dimension. Therefore, these
test functions with different features are used to testify the
effectiveness of the improved chicken swarm optimization
(ICSO) algorithm.
The comparison with the standard PSO [37], BA [7]
and the chicken swarm optimization [24] is also performed.
To be fair, the four algorithms run 50 times independently
for each function respectively, and the size of each swarm
is 50. The maximum number of iterations for each algorithm
is 1000. All benchmark functions are tested at D = 150.
Other parameters for four algorithms are listed in Table 4.
FIGURE 11. The average convergence curve for F2.
B. THE SIMULATION ANALYSIS
Four algorithms run independently 50 times for each test
function respectively, to obtain the best value, the worst value,
the mean value, the standard deviation and the mean run
time of 50 times. The experiment results obtained are shown
in Table 5. In addition, four algorithms run independently
FIGURE 12. The average convergence curve for F3.
50 times respectively for fourteen test functions at D = 150,
and the average convergence curves obtained as shown
in Figs. 10 to 23.
TABLE 4. The related parameter values for four algorithms.
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D. Wu et al.: Convergence Analysis and Improvement of the CSO Algorithm
TABLE 5. The comparison of the results of improved chicken swarm optimization and other algorithms for D = 150.
Boldface indicates a minimal value obtained for four
algorithms.
The best value and the average value may reflect
the convergence accuracy and optimization capabilities.
Table 5 shows that for the most standard test functions,
the improved chicken swarm optimization has a high conVOLUME 4, 2016
vergence accuracy, and is significantly better than the
other three algorithms. Especially for F1, F2, F4, F5,
F7, F8, F10, F11, F12, F13 and F14, the convergence
value obtained by the improved chicken swarm optimization algorithm are less than that of PSO, the chicken
swarm optimization and BA algorithm. Therefore, the the
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D. Wu et al.: Convergence Analysis and Improvement of the CSO Algorithm
FIGURE 13. The average convergence curve for F4.
FIGURE 17. The average convergence curve for F8.
FIGURE 14. The average convergence curve for F5.
FIGURE 18. The average convergence curve for F9.
FIGURE 15. The average convergence curve for F6.
FIGURE 19. The average convergence curve for F10.
FIGURE 16. The average convergence curve for F7.
FIGURE 20. The average convergence curve for F11.
chicken swarm optimization algorithm has a good advantage in processing multimodal and high-dimension complex
functions.
The worst value and the standard deviation can reflect the
robustness of the algorithm and the ability of confrontation
local optimum. From Table 5, we can see that for F6 and F7,
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For the optimization problems that are less requirement care
computation time, the improved chicken swarm optimization
algorithm has obvious advantages because of its high convergence accuracy.
As is shown in Figures 10 to 23, the convergence accuracy
of the improved chicken swarm optimization is better than
that of the chicken swarm optimization, and much better than
that of PSO and BA. As for convergence rates, the improved
chicken swarm optimization has higher convergence rates
compared with the chicken swarm optimization, PSO and BA
for most test functions.
FIGURE 21. The average convergence curve for F12.
FIGURE 22. The average convergence curve for F13.
VII. CONCLUSIONS AND FUTURE WORK
Based on the chicken swarm optimization, the state spaces of
the chicken and chicken swarm are described. The Markov
chain model of the chicken swarm optimization is established
by defining the chicken swarm state transition sequence, and
a detailed analysis of the properties of the Markov chain is
made. Note that it is a finite homogeneous Markov chain. The
final transfer status of the chicken swarm state sequence is
analyzed, and the global convergence of the chicken swarm
optimization is verified by the convergence criteria of a
random search algorithm. According to the problem that
the chicken swarm optimization is easy to fall into a local
optimum in solving high-dimensional problems, an improved
chicken swarm optimization is proposed. The relevant parameter analysis and the verification of the optimization capability by test functions in high-dimensional case are made.
In our future work, the in-depth study of the theory will
be conducted for the chicken swarm optimization, such as
designing hybrid chicken swarm optimization algorithm for
high-dimensional complex optimization problems and further studying convergence of the algorithm by martingale
sequence theory.
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FIGURE 23. The average convergence curve for F14.
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DINGHUI WU was born in Hefei, Anhui
Province, China. He received the Ph.D. degree in
control science and engineering from the School of
Internet of Things Engineering, Jiangnan University, Wuxi, China, in 2010. He was a Post-Doctoral
Fellow in the Textile Engineering in 2014. From
2014 to 2015, he was a Visiting Scholar with the
School of Computer and Electronic Engineering,
University of Denver, USA. He is currently an
Associate Professor and a Master Tutor with the
School of Internet of Things Engineering, Jiangnan University. His current
research interests include wind power generation technology, robot technology, embedded systems, and other aspects of the research and the teaching
of power electronics.
SHIPENG XU was born in Pingdingshan, Henan
Province, China. He received the B.Sc. degree
from the Zhengzhou University of Light Industry,
Zhengzhou, China, in 2014. He is currently pursuing the M.Sc. degree with the School of Internet
of Things Engineering, Jiangnan University, Wuxi,
China. His interests include intelligent optimization and shop scheduling.
FEI KONG was born in Hefei, Anhui Province,
China. He received the M.Sc. degree from
the School of Internet of Things Engineering,
Jiangnan University, Wuxi, China, in 2016.
His interests include intelligent optimization and
shop scheduling.
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