Diversity Handling In Evolutionary Landscape
Maumita Bhattacharya
School of Computing & Mathematics
Charles Sturt University, Australia
mbhattacharya@csu.edu.au
Abstract
The search ability of an Evolutionary Algorithm
(EA) depends on the variation among the individuals in
the population [3, 4, 8]. Maintaining an optimal level
of diversity in the EA population is imperative to
ensure that progress of the EA search is unhindered by
premature convergence to suboptimal solutions.
Clearer understanding of the concept of population
diversity, in the context of evolutionary search and
premature convergence in particular, is the key to
designing efficient EAs. To this end, this paper first
presents a comprehensive analysis of the EA
population diversity issues. Next we present an
investigation on a counter-niching EA technique [4]
that introduces and maintains constructive diversity in
the population. The proposed approach uses informed
genetic operations to reach promising, but un-explored
or under-explored areas of the search space, while
discouraging premature local convergence. Simulation
runs on a number of standard benchmark test functions
with Genetic Algorithm (GA) implementation shows
promising results.

1. Introduction
Implementation of EA requires preserving a
population while converging to a solution, which
maintains a degree of population diversity [7, 8, 9, 12,
13, 14, 15, and 16] to avoid premature convergence to
sub-optimal solutions. It is difficult to precisely
characterize the possible extent of premature
convergence as it may occur in EA due to various
reasons. The primary causes are algorithmic features
like high selection pressure and very high gene flow
among population members. Selection pressure pushes
the evolutionary process to focus more and more on the
already discovered better performing regions or
“peaks” in the search space and as a result population
diversity declines, gradually reaching a homogeneous
state. On the other hand unrestricted recombination
results in high gene flow which spreads genetic
material across the population pushing it to a
homogeneous state. Variation introduced through

mutation is unlikely to be adequate to escape local
optimum or optima [17]. While premature
convergence [17] may be defined as the phenomenon
of convergence to sub-optimal solutions, geneconvergence means loss of diversity in the process of
evolution. Though, the convergence to a local or to the
global optimum cannot necessarily be concluded from
gene convergence, maintaining a certain degree of
diversity is widely believed to help avoid entrapment
in non-optimal solutions [3, 4].
In this paper we present a comprehensive analysis
on population diversity in the context of efficiency of
evolutionary search. We then present an investigation
on a counter niching-based evolutionary algorithm that
aims at combating gene-convergence (and premature
convergence in turn) by employing intelligent
introduction of constructive diversity [4].
The rest of the paper is organized as follows:
Section 2 presents an analysis of diversity issues and
the EA search process; Section 3 introduces the
problem space for our proposed algorithm. Sections 4,
5 and 6 present the proposed algorithm, simulation
details and discussions on the results respectively.
Finally, Section 7 presents some concluding remarks.

2. Population diversity and evolutionary
search
In the context of EA, diversity may be described as
the variation in the genetic material among individuals
or candidate solutions in the EA population. This in
turn may also mean variation in the fitness value of the
individuals in the population. Two major roles played
by population diversity in EA are: Firstly, diversity
promotes exploration of the solution space to locate a
single good solution by delaying convergence;
secondly, diversity helps to locate multiple optima
when more than one solution is present [8, 15 and 16].
Besides the role of diversity regarding premature
convergence in static optimization problems, diversity
also seems to be beneficial in non-stationary
environments. If the genetic material in the population
is too similar, i.e., has converged towards single points

in the search space, all future individuals will be
trapped at that single point even though the optimal
solution has moved to another location in the fitness
landscape. However, if the population is diverse, the
mechanism of recombination will continue to generate
new candidate solutions making it possible for the EA
to discover new optima.
The search ability of a typical evolutionary
algorithm depends on the variation among the
individuals or candidate solutions in the population.
The variation is introduced by the recombination
operator to recombine existing solutions, and the
mutation operator introducing noise by applying
random variation to the individual's genome. However,
as the algorithm progresses, loss of diversity or loss of
genetic variation in the population results in low
exploration, pushing the algorithm to converge
prematurely to a local optimum or non-optimal
solution. The following sub-section presents an
analysis of the impact of population diversity on
premature convergence based on the concepts
presented in [13].

2.1 Effect of population diversity on premature
convergence
Let

r 
X =  X1,K, X
N



 ∈ S N


be a population of

individuals Y in the solution space S N , where the
r
population size is N ; X (0) be the initial population;
H is a schema, i.e., a hyperplane of the solution space
S . H may be represented by its defining components
(defining alleles) and their corresponding values as
H ai1,K , aik , where K (1 ≤ K ≤ chromosome length ) .

(

)

Leung et al. in [13] have proposed the following
measures related to population diversity in canonical
genetic algorithm.
r
Degree of population diversity, δ X : Defined as
the number of distinct components in the vector
N
r
∑ X i ; and Degree of population maturity, µ X :
i =1
r
r
Described as µ X = l − δ X or the number of lost
alleles.
With probability of mutation, p (m ) = 0 and
r
r
X (0 ) = X 0 , according to Leung [13] the following
postulates
hold
true:
For
each
solution,


r
Y ∈ H ai1,K, a  r  ; X 0  , there exists a n ≥ 0


iµ  X 0 





( )

( )

( )

( )

r
r
r
Probabilit y Y ∈ X (n ) / X (0 ) = X 0 > 0 .
Conversely,
for
each
solution,

r 
Y ∉ H ai1 ,K, a  r  ; X 0  , and every n ≥ 0 such


iµ  X 0 




r
r
r
that Probabilit y Y ∈ X (n ) / X (0 ) = X 0 = 0 .

such

{

that

{

}

}

It is obvious from the above postulates that a
canonical genetic algorithm’s search ability is confined
r
δ  X 
to the minimum schema with 2
different
individuals. Hence, greater the degree of population
r
diversity, µ X , greater is the search ability of the
genetic algorithm. Conversely, a small degree of
population diversity will mean limited search ability,
r
reducing to zero search ability with µ X = 0 .

( )

( )

2.2 Enhanced EAs to combat diversity issues
No mechanism in a standard EA guarantees that the
population will remain diverse throughout the run [17].
Although there is a wide coverage of the fitness
landscape at initialization due to the random
initialization of individuals’ genomes, selection
quickly eliminates the least fit solutions, which implies
that the population will converge towards similar
points or even single points in the search space. Since
the standard EA has limitations to maintain population
diversity, several models have been proposed by the
EA community which either maintain or reintroduce
diversity in the EA population [1, 2, 4, 5, 6, 10, 11, 14
and 18]. The key researches can be broadly categorized
as follows [15]:
a) Complex population structures to control gene
flow, e.g., the diffusion model, the island model,
the multinational EA and the religion model.
b) Specialized operators to control and assist the
selection procedure, e.g., crowding, deterministic
crowding, and sharing are believed to maintain
diversity in the population.
c) Reintroduction of genetic material, e.g., random
immigrants and mass extinction models are aimed
at reintroduction of diversity in the population.
d) Dynamic Parameter Encoding (DPE), which
dynamically resizes the available range of each
parameter by expanding or reducing the search
window.
e) Diversity guided or controlled genetic algorithms
that use a diversity measure to assess and control
the survival probability of individuals and the
process of exploration and exploitation.

Figure 1 summarizes the major methods proposed
to directly or indirectly control EA population
diversity.
Manual tuning of
constants/functions
Measure based
control
Parameter control
based methods

Self-adaptive
control
Polulation structure
based control

In the second case, i.e., to locate multiple good
solutions, alternative and different solutions may have
to be considered before accepting one final solution as
the optimum. An algorithm that can keep track of
multiple optima simultaneously should be able to find
multiple optima in the same run by spreading out the
search. On the other hand, maintaining genetic
diversity in the population can be primarily beneficial
in the first case; the problem of entrapment in local
optima.

Crowding
Replacement schemes
Deterministic and
probabilistic crowding

Remarks: The issue is - how much genetic diversity
in the population is optimum?

Cellular EA
Spatial population
topologies

Patch work model
Religion based EA

Selection based
techniques

Sharing
Diversity control
based EA

Direct/indirect control
of
population diversity

Forking EA
Search space division

Shifting balace EA
Multinational EA
Random immigrants EA

Mass exinction
Extinction model
SOC extinction model
Restart and phase based
techniques

CHC algorithm
Diversity guided EA

Fig.1. Direct/indirect control of population diversity in EA.

3. Understanding the problem space
Before we present our proposed approach, which
aims at achieving constructive diversity, it is important
to understand the problem space we are dealing with.
For optimization problems the main challenge is often
posed by the topology of the fitness landscape, in
particular its ruggedness in terms of local optima. The
target optimization problems for our approach are
primarily multimodal. Genetic diversity of the
population is particularly important in case of
multimodal fitness landscape. Evolutionary algorithms
are required to avoid and escape local optima or basins
of attraction to reach the optimum in a multimodal
fitness landscape.
Over the years, several new and enhanced EAs have
been suggested to improve performance [1, 2, 4, 5, 6,
10, 11 and 14]. The objectives of such research are
twofold; firstly, to avoid stagnation in local optimum in
order to find the global optimum; secondly, to locate
multiple good solutions if the application requires so.

Recombination in a fully converged population
cannot produce solutions that are different from the
parents; leave alone better than the parents. However, a
very high diversity actually deteriorates performance
of the recombination operator. Offspring generated
combining two parents approaching two different
peaks is likely to be placed somewhere between the
two peaks; hindering the search process from reaching
either of the peaks. This makes the recombination
operator less efficient for fine-tuning the solutions to
converge at the end of the run. Hence, the optimal level
of diversity is somewhere between fully converged and
highly diverse. Various diversity measures (such as
Euclidean distance among candidate solutions, fitness
distance and so on) may be used to analyze algorithms
to evaluate their diversity maintaining capabilities.
In the following sections we investigate the
functioning and performance of our proposed Counter
Niching-based Evolutionary Algorithm [4].

4. Counter Niching EA: Landscape
information, informed operator and
constructive diversity
To attain the objective of introducing constructive
diversity in the population, the proposed technique first
extracts information about the population landscape
before deciding on introduction of diversity through
informed mutation. The aim is to identify locally
converging regions or donor communities in the
landscape whose redundant less fit members (or
individuals) could be replaced by more promising
members sampled in un-explored or under-explored
sections of the decision space. The existence of such
communities is purely based on the position and spread
of individuals in the decision space at a given point in
time. Once such regions are identified, random
sampling is done on yet to be explored sections of the
landscape. Best representatives found during such
sampling, now replace the worst members of the

identified donor regions. Best representatives are the
ones that are fitness wise the fittest and spatially the
farthest. Here, average Euclidean distance from
representatives of all already considered regions
(stored in a memory array) is the measure for spatial
distance. Regular mutation and recombination takes
place in the population as a whole. The basic
framework is as depicted in Figure 2.
The Counter Niching based EA
Identify Donor Regions and perform
informed operation
Find Communities or
Clusters in population
(Procedure GRID_NICHING)
(Procedure INFORMED_OP)

Fig.2. The COUNTER-NICHING based EA framework

The task described above is carried out by the
following three procedures:
Procedure COUNTER NICHING EA: This is the main
algorithm.
Procedure GRID NICHING: This procedure is called
by COUNTER NICHING EA and is used to identify
tendency of genotypic convergence.
Procedure INFORMED OP: This procedure is second
in order to be called by COUNTER NICHING EA.
This procedure is responsible of genetic operations
including informed mutation.
Figure
3
presents
the
procedure
COUNTER_NICHING_EA. For details on the
procedures GRID NICHING and INFORMED OP, we
refer to our previous work on [4].

5. Simulations
5.1 Test functions
The proposed algorithm is tested on a set of
commonly used benchmark test functions to validate
its efficacy. The standard benchmark test function set
used in the simulation runs consists of minimization of
seven analytical functions as follows: Ackley’s Path
Function ( f ack ( x ) ), Griewank’s Function f gri ( x ) ,
Rastrigin’s

Function

f rtg ( x ) ,

Generalized

Rosenbrock’s function f ros (x ) , Axis parallel HyperEllipsoidal Function or Weighted Sphere Model
f elp (x ) , Schwefel Function 1.2 f sch − 1.2 (x ) and a
rotated Rastrigin Function f rrtg (x ) .
Algorithm 1: Procedure COUNTER NICHING EA
1: begin
2: t = 0
3: Initialize population P(t)
4: Evaluate population P(t)
5: while (not<termination condition>)
6: begin
t=t+1
7:
(* Perform pseudo-niching of the population*)
8: Call Procedure GRID_NICHING
(* Perform informed genetic operations *)
9: Call Procedure INFORMED_OP
10: Create new population using an elitist selection
mechanism
11: Evaluate P(t)
14: end while
15: end
Fig.3. The COUNTER-NICHING based EA framework.

5.2 Algorithms considered for comparison
The algorithms used in the comparison are as
follows: (a) the “standard EA” (SEA), (b) the self
organized criticality EA (SOCEA), (c) the cellular EA
(CEA), and (d) the diversity guided EA (DGEA). The
SEA uses Gaussian mutation with zero mean and
variance σ 2 = 1 + t + 1 . The SOCEA is a standard
EA with non-fixed and non-decreasing variance
σ 2 = POW (10 ) , where POW (α ) is the power-law
distribution. The CEA uses a 20x20 grid with wrapped
edges. The grid size corresponds to the 400 individuals
used in the other algorithms. The CEA uses Gaussian
mutation with variance σ 2 = POW (10 ) . Finally, the
DGEA uses the Gaussian mutation operator with
variance σ 2 = POW (1) . The reader is referred to [15]
for additional information on the methods.

5.3 Experiment set-up
Simulations were carried out to apply the proposed
COUNTER NICHING based EA with real-valued
encoding with parameters N (population size) =300,

p m (mutation
probability)
=0.01
and
pr
(recombination probability) =0.9. In case of the
algorithms used for comparison as mentioned in
Section 4.2, namely, (i) SEA (Standard EA), (ii)
SOCEA (Self-organized criticality EA), (iii) CEA (The
Cellular EA), and (iv) DGEA (Diversity guided EA),
experiments were performed using real-valued
encoding, a population size of 400 individuals, binary
tournament selection. Probability of mutating an entire
genome was p m = 0.75 and probability for crossover
was p r = 0.9. All the test functions were considered in
20, 50 and 100 dimensions. Reported results were
averaged over 30 independent runs, maximum number
of generations in each run being only 500, as against
1000 generations in used [15] for the same set of test
cases for the 20 dimensional scenarios. The
comparison algorithms use 50 times the dimensionality
of the test problems as the terminating generation
number in general, while the COUNTER NICHING
EA uses 500, 1000 and 2000 generations for the 20, 50
and 100 dimensional problem variants respectively.

6. Results and discussions
This section presents the empirical results obtained
by the COUNTER NICHING EA algorithm when
tackling the seven test problems mentioned in Section
5.1 with dimensions 20, 50 and 100.

6.1 General performance of COUNTER
NICHING EA
Table 1 presents the error values, ( f (x ) − f (x )* )
where,
f (x )* is the optimum. Each column
corresponds to a test function. The error values have
been presented for the three dimensions of the
problems considered, namely 20, 50 and 100. As each
test problem was simulated over 30 independent runs,
we have recorded results from each run and sorted the
results in ascending order. Table 1 presents results
from the representative runs: 1st (Best), 7th, 15th
(Median), 22nd and 30th (Worst), Mean and Standard
Deviation (Std). The main performance measures used
are the following:
“A” Performance: Mean performance or average
of the best-fitness function found at the end of each
run. (Represented as ‘Mean’ in Table 1).
“SD”
Performance:
Standard
deviation
performance. (Represented as ‘Std.’ in Table 1).

“B” Performance: Best of the fitness values
averaged as mean performance. (Represented as ‘Best’
in Table 1).
As can be observed COUNTER NICHING EA has
demonstrated descent performance in majority of the
test cases. However, as can be seen from the
highlighted segment (highlighted in bold) of Table 1,
the proposed algorithm was not very efficient in
handling the higher dimensional cases (50 and 100
dimensional cases in this example) for the rotated
Rastrigin Function f rrtg (x ) . Keeping in mind the

concept of No Free Lunch Theorem, this is acceptable
as no single algorithm can be expected to perform
favorably for all possible test cases. The chosen
benchmark test functions represent a wide variety of
test cases.

6.2 Comparative performance of COUNTER
NICHING EA
Simulation results obtained with COUNTER
NICHING EA in comparison to SEA, SOCEA, CEA,
and DGEA (see Section 5.2 for descriptions of these
algorithms) are presented in Table 2. Results reported
in this case, for COUNTER NICHING EA were
averaged over 50 independent runs. These simulation
results ascertain COUNTER NICHING EA’s superior
performance as regards to solution precision in all the
test cases particularly for lower dimensional instances.
This may be attributed to COUNTER NICHING EA’s
ability to strike a better balance between exploration
and exploitation. However, the proposed algorithm’s
performance deteriorates with increasing dimensions.
Also, the algorithm could not handle the high
dimensional versions of the high epistatis rotated
Rastrigin function to any satisfactory level. For the
reported results as shown in Table 2, the 100
dimensional scenarios of the test problems used 5000
generations for each of the compared algorithm,
namely, SEA, SOCEA, CEA and DGEA. On the other
hand, COUNTER NICHING EA used only 2000
generations to reach the reported results.

6.3 An analysis of population diversity for
COUNTER NICHING EA
In the next phase of our experiments, we have
investigated COUNTER NICHING EA’s performance
in terms of maintaining constructive diversity. There
are various measures of diversity available. The
“distance-to-average-point” measure used in [15] is
relatively robust with respect to population size,
dimensionality of problem and the search range of each

Table 1. Error Values Achieved on the Test Functions with
Simulation Runs for COUNTER NICHING EA. Dimensions
of Each Function Considered are 20, 50 and 100.
f ack f
f
f
f
f
f

gri

1st
(Be
st)
7th

2
0
D

15th
(M
edi
an)
22n
d

30th
(W
orst
)
Me
an

Std.
1st
(Be
st)
7th

5
0
D

15th
(M
edi
an)
22n
d

30th
(W
orst
)
Me
an
Std.
1st
(Be
st)
7th
1
0
0
D

15th
(M
edi
an)
22n
d

1.00
E61
1.11
E61
1.11
E61

4.3E
-62

rtg

ros

elp

sch −

rrtg

1.01
E60
1.01
E60
1.11
E60

2.01
E60
2.01
E60
2.89
E60

2.01E50

3.89
E-6

4.41
E62
4.96
E62

1.11
E61
1.13
1E61
1.21
0E61

2.01E50

3.89
E-6

2.71E50

3.9E
-6

1.95
E61
3.11
E61

5.61
E62
8.71
E62

2.02
E61
3.30
E61

1.92
E60
2.01
E60

2.91
E60
2.91
E60

2.91E50

3.93
E-6

2.91E50

3.99
E-6

1.12
E61

5.02
E62

1.21
E61

1.12
E60

2.92
E60

2.72E50

3.9E
-6

8.33
E62
0.56
E29
0.71
E29
0.71
E29

1.64
E62
1.00
E30
1.01
E30
1.01
E30

8.6E
-62

4.62
E61
1.01
E30
1.01
E30
1.10
E30

4.22E51

3.88
-8

1.00
E30
1.01
E30
1.10
E30

4.71
E61
1.21
E29
1.41
E29
1.90
E29

2.21E20

9.01

2.40E20

9.01

2.90E20

9.11

0.91
E29
0.99
E29

1.91
E30
1.99
E30

1.81
E30
1.99
E30

1.92
E29
1.98
E29

1.51
E30
1.92
E30

2.91E20

9.22

2.91E20

9.24

0.73
E29
1.55
E30

1.11
E30
4.41
E31
1.90
E-9

1.91
E29
3.16
E30
2.09
E-9

1.11
E30
3.66
E31
2.09
E-8

2.90E20

9.12

3.16E21

0.09
8

1.00
E-9

1.11
E30
4.76
E31
1.20
E-9

2.09E5

10.5
2

1.01
E-9

1.51
E-9

1.92
E-9

2.91
E-9

2.92
E-8

2.59E5

10.6
6

1.12
E-9

1.72
E-9

1.99
E-9

2.99
E-9

2.99
E-8

3.29E5

11.0
9

1.36
E-9

1.86
E-9

2.21
E-9

3.21
E-9

3.21
E-8

3.79E5

11.6
1

30th
(W
orst
)
Me
an

1.36
E-9

1.92
E-9

2.92
E-9

3.92
E-9

3.90
E-8

3.98E5

11.7
9

1.13
E-9

1.81
E-9

2.01
E-9

3.03
E-9

3.01
E-8

3.69E5

11.5
0

Std.

1.61
E10

2.71
E10

3.88
E10

6.91
E10

5.81
E-9

7.48E6

0.52
41

variable. Hence, we have used this measure of
diversity in our investigation. The “distance-toaverage-point” measure for N dimensional numerical
problems can be described as below [15].
P
2
N
1
diversity (P ) =
⋅ ∑
∑  sij − s j  (1)

L ⋅ P i = 1 j = 1
where, L is the length of the diagonal or range in
the search space S ⊆ ℜ N , P is the population, P is
the population size, N is the dimensionality of the
problem, sij is the j ’th value of the i ’th individual,
and s j is the

j ’th value of the average point s . It is

assumed that each search variable

s k is in a finite

range, s k _ min ≤ s k ≤ s k _ max . Table 3 depicts the
average diversity for four test problems with
COUNTER NICHING EA simulation runs. The values
reported in Table 3, averages the value of the diversity
measure in equation (1) calculated at each generation
where there has been an improvement in average
fitness over 500, 1000 and 2000 generations for the 20,
50 and 100 dimensional cases respectively. Final
values were averaged over 100 runs. To eliminate the
noise in the initial generations of a run, diversity
calculation does not start until the generation, since
which a relatively steady improvement in fitness has
been observed. Table 3 shows that the COUNTER
NICHING EA does not necessarily maintain very high
average population diversity. However, EA’s
requirement is not to maintain very high average
population diversity but to maintain an optimal level of
population diversity. The high solution accuracy
obtained by COUNTER NICHING EA proves that the
algorithm is successful in this respect.

6.4 Statistical significance of comparative
analysis
Finally, a t-test (at 0.05 level of significance; 95%
confidence) was applied in order to ascertain if
differences in the “A” performance for the best average

Table 2. Average Fitness Comparison for SEA, SOCEA, The
CEA, DGEA, and COUNTER NICHING EA*. Dimensions of
Each Function considered are 20, 50 and 100. ‘-’ Appears
Where the Corresponding Data is Not Available.
Function

f ack ( x )

SEA

SOCEA

CEA

DGEA

C_EA*

2.494

0.633

0.239

3.36E-5

1.08E61

1.171

0.930

0.642

7.88E-8

4.6E-62

11.12

2.875

1.250

3.37E-8

1.21E61

8292.3
2

406.490

149.05
6

8.127

1.0E-60

-

-

-

-

2.9E-60

20D

f gri (x )
20D

f rtg (x )
20D

f ros (x )
20D

f elp (x )
20D

f sch − 1.2 ( -

-

-

-

2.7E-50

-

-

-

-

3.9E-6

2.870

1.525

0.651

2.52E-4

1.01E29

1.616

1.147

1.032

1.19E-3

1.01E30

44.674

22.460

14.224

1.97E-6

2.01E30

41425.
674

4783.246

1160.0
78

59.789

1.91E29

-

-

-

-

1.00E30

f sch − 1.2 ( -

-

-

-

2.9E-20

-

-

-

-

9.1

2.893

2.220

1.140

9.80E-4

1.00E-9

2.250

1.629

1.179

3.24E-3

1.80E-9

106.21
2

86.364

58.380

6.56E-5

2.00E-9

91251.
300

30427.63

6053.8
70

880.324

3.00E-9

-

-

-

-

2.99E-8

f sch − 1.2 ( -

-

-

-

3.7E-5

-

-

-

11.51

20D

f rrtg (x )
20D

f ack ( x )
50D

f gri (x )
50D

f rtg (x )
50D

f ros (x )
50D

f elp (x )
50D
50D

f rrtg (x )
50D

f ack ( x )
100D

f gri (x )
100D

f rtg (x )
100D

f ros (x )
100D

f elp (x )
100D
100D

f rrtg (x )
100D

-

fitness function are statistically significant when
compared with the one for the other techniques used
for comparison. The P -values of the two-tailed t-test
are given in Table 4. As can be observed, the
difference in “A” performance of COUNTER
NICHING EA is statistically significant for majority of
the techniques across the test functions in their three
different dimensional versions.
Table 3. Average Population Diversity Comparison For
COUNTER NICHING EA (Fixed Run). An average of 100
Runs Have Been Reported In Each Case.
20D

50D

100D

f ack ( x )

0.001350

0.001811

0.002001

f gri (x )

0.001290

0.001725

0.002099

f rtg (x )

0.003000

0.003550

0.004015

f ros (x )

0.001718

0.002025

0.002989

7. Conclusions
In this paper we investigated the issues related to
population diversity in the context of the evolutionary
search process. We established the association between
population diversity and the search ability of a typical
evolutionary algorithm. Then we presented an
investigation on an intelligent mutation based EA that
tries to achieve optimal diversity in the search
landscape. The framework basically incorporates two
key processes.
Firstly, the population’s spatial
information is obtained with a pseudo-niching
algorithm. Secondly, the information is used to identify
potential local convergence and community
formations. Then diversity is intelligently introduced
with informed genetic operations, aiming at two
objectives: (a) Promising samples from unexplored
regions are introduced replacing redundant less fit
members of over-populated communities. (b) While
local entrapment is discouraged, representative
members are still preserved to encourage exploitation.
While the current focus of the research was to
introduce and maintain population diversity to avoid
local entrapment, this Counter Niching-based
algorithm can also be adapted to serve as an
inexpensive alternative for niching GA, to identify
multiple solutions in multimodal problems as well as to
suit the diversity requirements of a dynamic
environment.

Table 4. The P -values of the t-test with 99degrees of
freedom. Dimensions of Each Function considered are 20, 50
and 100. ‘-’ Appears Where the Corresponding Data is Not
Available.
C_EA*SEA

C_EA*SOCEA

C_EA*CEA

C_EA*DGEA

f ack (x ) 20D

0.1144

0.4263

0.625

0.9954

f gri (x ) 20D

0.2793

0.3349

0.4231

0.9998

f rtg (x ) 20D

0.0009

0.0901

0.2636

0.9999

f ros (x ) 20D

0

0

0

0.0044

f ack ( x ) 50D

0.0903

0.217

0.4198

0.9873

f gri (x ) 50D

0.2037

0.2843

0.3098

0.9725

f rtg (x ) 50D

0

0

0.0002

0.9989

f ros (x ) 50D

0

0

0

0

f ack ( x ) 100D

0.0891

0.1363

0.2857

0.975

f gri (x ) 100D

0.1337

0.2019

0.2776

0.9546

f rtg (x ) 100D

0

0

0

0

f ros (x ) 100D

0

0

0

0

Function

References
[1] ADRA, S. F. AND FLEMING, P. J. 2011. Diversity
management in evolutionary many-objective
optimization. IEEE Trans. Evol. Comput. 15, 2, 183–
195.
[2] ARAUJO, L. AND MERELO, J. J. 2011. Diversity
through multiculturality: Assessing migrant choice
policies in an island model. IEEE Trans. Evol.
Comput. 15, 4, 456–468.
[3] Bhattacharya, M., “An Informed Operator Approach
to Tackle Diversity Constraints in Evolutionary
Search”, Proceedings of The International
Conference on Information Technology, ITCC 2004,
Vol. 2, IEEE Computer Society Press, ISBN 0-76952108-8,pp. 326-330.
[4] Bhattacharya, M., “Counter-niching for Constructive
Population Diversity”, in Proceedings of the 2008
IEEE Congress on Evolutionary Computation (CEC
2008), Hong Kong, IEEE Press, ISBN: 978-1-42441823-7, pp. 4174-4179.
[5] CHOW, C. K. AND YUEN, S. Y. 2011. An
evolutionary algorithm that makes decision based on
the entire previous search history. IEEE Trans. Evol.
Comput. 15, 6, 741–769.

[6] CURRAN, D. AND O’RIORDAN, C. 2006.
Increasing population diversity through cultural
learning. Adapt. Behav. 14, 4, 315–338.
[7] FRIEDRICH, T., HEBBINGHAUS, N., AND
NEUMANN, F. 2007. Rigorous analyses of simple
diversity mechanisms. In Proceedings of the Genetic
and Evolutionary Computation Conference. 1219–
1225.
[8] FRIEDRICH, T., OLIVETO, P. S., SUDHOLT, D.,
AND WITT, C. 2008. Theoretical analysis of
diversity mechanisms for global exploration. In
Proceedings of the Genetic and Evolutionary
Computation Conference. 945–952.
[9] GALV´AN-L´OPEZ,
E.,MCDERMOTT,
J.,
O’NEILL, M., AND BRABAZON, A. 2010.
Towards an understanding of locality in genetic
programming. In Proceedings of the 12th Annual
Conference
on
Genetic
and
Evolutionary
Computation. 901–908.
[10] GAO, H. AND XU, W. 2011. Particle swarm
algorithm with hybrid mutation strategy. Appl. Soft
Comput. 11, 8, 5129–5142.
[11] JIA, D., ZHENG, G., AND KHAN, M. K. 2011. An
effective memetic differential evolution algorithm
based on chaotic local search. Inform. Sci. 181, 15,
3175–3187.
[12] K. A. De Jong, “An Analysis of the Behavior of a
Class of Genetic Adaptive Systems”, PhD thesis,
University of Michigan, Ann Arbor, MI, Dissertation
Abstracts International 36(10), 5140B, University
Microfilms Number 76-9381, 1975.
[13] Leung, Y., Gao, Y. and Xu, Z. B., "Degree of
Population Diversity-A Perspective on Premature
Convergence in Genetic Algorithms and its Markov
Chain Analysis",IEEE Transactions on Neural
Networks, volume 8,no. 5, pp. 1165-1176,1997.
[14] LIANG, Y. AND LEUNG, K.-S. 2011. Genetic
algorithm with adaptive elitist-population strategies
for multimodal function optimization. Appl. Soft
Comput. 11, 2, 2017–2034.
[15] R. K. Ursem, “Diversity-Guided Evolutionary
Algorithms”, Proceedings of Parallel Problem
Solving from Nature VII (PPSN-2002), 2002, pp.
462-471.
[16] R. Thomsen and P. Rickers, “Introducing Spatial
Agent-Based Models and Self-Organised Criticality
to Evolutionary Algorithms” Master’s thesis,
University of Aarhus, Denmark, 2000.
[17] T. B¨ack, D. B. Fogel, Z. Michalewicz, and others,
(eds.), Handbook on Evolutionary Computation, IOP
Publishing Ltd and Oxford University Press, 1997.
[18] Bhattacharya, Maumita. "Meta Model Based EA for
Complex Optimization." International Journal of
Computational Intelligence 4.1 (2008).

