CULTURAL SWARMS

Knowledge-driven Framework for Solving Nonlinearly Constrained Global

Optimization Problems

Mostafa Z. Ali, Yaser Khamayseh

Computer Information Systems, Jordan University of Science & Technology, Irbid, Jordan

Robert G. Reynolds

Department of Computer Science, Wayne State University, Detroit, Michigan, U.S.A

Keywords: Evolutionary computation, Nonlinearly constrained global optimization problem, Cultural swarms, Social

interaction, Knowledge source interaction.

Abstract: In this paper we investigate how diverse knowledge sources interact to direct individuals in a swarm

population influenced by a social fabric approach to efficiently solve nonlinearly constrained global

minimization problems. We identify how knowledge sources used by Cultural Algorithms are combined to

direct the decisions of the individual agents in solving optimization problems using an influence function

family based upon a Social Fabric metaphor. The interaction of these knowledge sources with the

population swarms produced emergent phases of problem solving. This reflected an algorithmic process that

emerged from the interaction of the knowledge sources under the influence of a social fabric using different

configurations. This suggests that the social interaction of individuals coupled with their interaction with a

culture within which they are embedded provides a powerful vehicle for the solution of nonlinearly

constrained optimization problems. The algorithm can escape from the previously converged local

minimizers, and can converge to an approximate global minimizer of the problem asymptotically.

Numerical experiments show that it is better than many other well-known recent methods for constrained

global optimization.

1 INTRODUCTION

The Cultural Algorithm (CA) is a class of

computational models derived from observing the

cultural evolution process in nature. It is a dual

inheritance system that characterizes evolution in

human culture at both the macro-evolutionary level,

which takes place within the belief space, and at the

micro-evolutionary level, which occurs at the

population space. Knowledge produced in the

population space at the micro-evolutionary level is

selectively accepted or passed to the belief space and

used to adjust the symbolic structures there. This

knowledge can then be used to influence the changes

made by the population in the next generation. The

basic framework is shown in Figure 1.

Previous work by Reynolds (Reynolds and

Saleem, 2003) identified five basic categories of

knowledge that were useful in decision making.

They were normative knowledge (ranges of

acceptable behaviours), situational knowledge

(exemplars of successful and unsuccessful

solutions), domain knowledge (knowledge of

domain objects, their relationships, and interactions),

history knowledge (temporal patterns of behaviour),

and topographical knowledge (spatial patterns of

behaviour). This set of categories is viewed as being

complete for a given domain in the sense that all

available knowledge can be expressed in terms of

one of these classifications.

Reynolds (Reynolds and Saleem, 2003) looked at

the roles and contribution of these five generic

knowledge classes (normative, topographical,

domain, situational, and history knowledge) to the

optimization problem-solving process using

Evolutionary Programming (EP) as the population

model. They observed the emergence of certain

103

Z. Ali M., Khamayseh Y. and G. Reynolds R. (2009).

CULTURAL SWARMS - Knowledge-driven Framework for Solving Nonlinearly Constrained Global Optimization Problems.

In Proceedings of the International Joint Conference on Computational Intelligence, pages 103-110

DOI: 10.5220/0002282301030110

 SciTePress

problem solving phases in terms of the relative

performance of different knowledge sources over

time. They labelled these phases as the coarse

grained, fine grained, and backtracking phases. Each

phase is characterized by the dominance of a suite or

subset of the knowledge sources that are most

successful in generating new solutions in that phase.

In fact, the dominant subset of knowledge sources is

often applied in a specific sequence within each

phase. It appears that one knowledge source

produces new solutions that are consequently

exploited in by another knowledge source.

Transitions between phases occur when the solutions

produced by one phase can be better exploited by

knowledge sources associated with the next phase.

These phases emerged in static, dynamic, and

deceptive problem environments.

Figure 1: The framework of cultural algorithm.

The coarse grained phase often dominates at the

beginning of the search process or when the problem

solving landscape changes dynamically, and a search

for a new solution must begin anew. In the coarse

grained phase topographical knowledge dominates,

producing the best new solution over 50% of the

time. Situational knowledge is the second most

successful, producing the best new solution over

25% of the time. In the fine-grained phase

situational knowledge is the most successful at

generating the best new individual, while Normative

and Domain knowledge are a distant second best. In

the backtracking phase all of the knowledge sources

are equally successful at generating new solution.

Static problems are the exception, in which cases the

history component has little effect. Likewise, in non-

deceptive environments backtracking occurred less

frequently than the other two phases.

Cultural Algorithms can provide a flexible

framework in which to study the emergence of

complexity in a multi-agent system (MAS)

(Reynolds, 1986). In this scenario the Cultural

Algorithms framework has been embedded with the

recursive porous agent simulation tool (Repast)

(North, Collier, and Vos, 2006), producing a toolkit

that is called Cultural Algorithms Toolkit (CAT).

This tool is used to view the power Cultural

Algorithms in solving many Engineering problems

and other type of problems (Reynolds and Ali

2007).

While many successful real-world applications

of Cultural Algorithms have been produced, we are

interested in studying the fundamental

computational processes involved the use of Cultural

Systems as problem solvers. In previous work the

influence of the knowledge sources have been on

individuals in the population only. The goal of this

paper is to examine how Cultural Algorithms solve

nonlinearly constrained global optimization

problems. In our investigation here, we employ a set

of standard test problems with differentiable

objective function. These test problems are

considered diverse enough to cover many kinds of

difficulties that constrained global optimization

faces. Agents then interact socially via the various

knowledge sources to find the optimum after

weaving the social fabric to motivate interaction. We

then investigate the emergence of social patterns in

both the population space and the belief space when

the problem is successfully solved.

In this new approach, the Social Fabric influence

function is the gear to find the optimal for a certain

minimization problem. The agents are connected

through a topology that determines connectivity type

between agents, through which the fabric is weaved

after the initial signal is sent from the Knowledge

Sources.

2 PREVIOUS WORK

Several researchers have used different types of

Algorithms for solving constrained optimization

problems. A quick overview is as follows:

Coello and Mezura (Coello and Mezura-Montes,

2002) implemented a version of the Niched-Pareto

Genetic Algorithm (NPGA) (Horn, Nafpliotis, and

Goldberg, 1994) to handle constraints in single-

objective optimization problems. The NPGA is a

multiobjective optimization approach in which

individuals are selected through a tournament based

on Pareto dominance. However, unlike the first

NPGA, Coello and Mezura’s approach does not

require niches (or fitness sharing (Deb and

Goldberg, 1989)) to maintain diversity in the

population. The NPGA is a more efficient technique

than traditional multiobjective optimization

IJCCI 2009 - International Joint Conference on Computational Intelligence

104

algorithms, because it only uses a sample of the

population to estimate Pareto dominance.

Deb (Deb and Goyal, 1996) proposed a Genetic

Adaptive Search (GeneAS) to solve engineering

optimization problems. He proposed to use both,

binary and real encoding for each solution. This

approach was tested on three engineering problems

(Deb and Goyal, 1996), making emphasis in

problems that have discrete and continuous

variables. The obvious drawback of the approach is

the need of implementing combined operators for

the special encoding adopted. Mezura-Montes

(Coello and Mezura-Montes, 2002) presented an

enhanced Evolutionary Algorithm that doesn’t

require the definition of extra parameters other than

those used by the Evolutionary strategy. The

implemented mechanism allows the algorithm to

maintain diversity during the process. Reynolds

(Reynolds and Peng, 2005) implemented an

algorithm that uses the Marginal Value Theorem

(MVT) to influence the individuals in the population

and drive the process of obtaining better solutions.

The algorithm was a more efficient one than the one

presented in (Coello and Mezura-Montes, 2002;

Coello, 2002) and (Coelho, Souza, and Mariani,

2009).

3 THE SOCIAL FABRIC

INFLUENCE FUNCTION

3.1 Concept

Knowledge sources are allowed to influence

individuals through a network. From a theoretical

perspective we view individuals in the real world as

participating in a variety of different networks.

Several layers of such networks can be supported

within a population. The interplay of these various

network computations is designated as the “social

fabric”. This notion of social fabric has appeared

metaphorically in various ways within Computer

Science. For example, IBM among others developed

tools to reinforce the “social fabric” whereby

designers and programmers interact to solve

complex problems (Cheng, Patterson, Rohall,

Hupfer, and Ross, 2005).

We adapt the Brock-Durlauf model of interactive

discrete choice (Brock and Durlauf ;2001) to

arbitrary interaction topologies represented by an

arbitrary adjacency matrix Г: All individuals faces

the binary choice set S = {-1, 1}: Let agent i choose

, ω

Є S, so as to maximize her utility, which

depends on the actions of her neighbours: Ui =

U(ω



,ω



), whereω



denotes the vector of

dimension d

containing as elements the decisions

made by each of agent i's neighbours, j . The

I-vector of all agents' decisions, ω = (ω



,…,ω



); is

also known as a configuration, and ω



is known

as agent i's environment. We assume that an agent's

utility function Ui is additively separable in a private

utility component, which without loss of generality

(due to the binary nature of the decision) may be

written as hω

, h > 0; in a social interactions

component, which is written in terms of quadratic

interactions between her own decision and of the

expectation of the decisions of her neighbours,ω



ω

















|



∑







; and a random utility

component, ω



; which is observable only by the

individual i.

The social fabric is viewed as a computational

tool that influences the action and interaction of the

various knowledge sources. Informally, we have N

networks and M individuals. An individual can be

associated with one or more networks. For a given

network only certain information is allowed to flow

along that network between nodes. Each network

can be viewed as being produced by a single thread

that links up the participating nodes.

3.2 Weaving the Social Fabric into the

CAT System

The networks that comprise the social fabric can

emanate from either the Belief Space or the

Population Space. In terms of the population, the

network could reflect a kinship network or an

economic network for example. In terms of the

Belief Space, the network could be the Internet, or a

local area network, or some other network directly

accessible to the knowledge sources. It may be that

the Knowledge Sources know something about the

networks that they can access but are not sure how

those networks are linked up to the low level social

networks of the population. In other words, they

may be aware of the outer layer of the social fabric,

but can only infer about what is in the interior lining.

The experimental framework for the social fabric

component is illustrated in Figure 2. The figure

shows the initialization step, where each individual

first will be affected by one knowledge source (as a

special case) that will represent the initial signal to

be passed to other individuals. The signal is passed

to adjacent individuals in the topology. The

individual is represented as a node in the landscape,

where the number of connections or hops over

CULTURAL SWARMS - Knowledge-driven Framework for Solving Nonlinearly Constrained Global Optimization

Problems

105

which it can transmit this information to its

neighbours will correspond to its influence, by a

maximum hop distance and will be limited. The

number of hops can be either 0 or d meaning either

no connections or d connections at a time. The

current system is using 0 hops as the individuals

don’t have any connections with each other.

From the standpoint of the Knowledge Sources

they can seed or influence a subset of the population,

and that subset may have population level affects but

they can only guess what they might be. The key is

to “seed” a subset of the population how represent

“the weave” to these other networks, assuming that

those that represent the weave between the networks

have certain properties.

Figure 2: Embedded Social Fabric component in CAT

with activated dynamics in the environment.

As a simple configuration in CAT we can simply

specify just one network, one that is accessible to the

Knowledge Sources in the Belief Space. What we

wish to investigate is whether just having access to

the Social Fabric is sufficient for the Knowledge

Sources to improve the performance of the influence

function as opposed to not having a network to

distribute their influence at all.

The process starts where each individual first

will be affected by one knowledge source (as a

special case) that will represent the initial signal to

be passed to other individuals. The signal is passed

to adjacent individuals in the topology based on the

network connectivity. The individual is represented

as a node in the landscape, where the number of

connections or hops over which it can transmit this

information to its neighbours corresponds to its

influence. The maximum number of hops can be

either 0 or d meaning either no connections or d

connections at a time. The simplest case is

configured by assuming that each individual is

connected to a fixed number of other individuals

using a constant topology. The topologies that we

used here were taken from work in Particle Swarm

Optimization where the impact of various topologies

on the communication of local information among

particles has been studied.

Several frequently used topologies taken from

the Particle Swarm Optimization literature are

supported in CAT. For example, the lBest model is

the simplest form of a local topology is known as the

ring the ring model. The lBest ring model connects

each individual to only two other individuals in the

landscape and is shown in figure 3(a). Another

frequently used topology is the gbest topology. In

this topology each individual in the network is

connected to all the individuals in the network as

shown in figure 3(b). The advantage of the lbest

model may lie in its lower convergence rate relative

to the gbest model which may reduce the change of

premature convergence to a false peak.

Another topology supported in CAT is the square

topology in which each individual has four

connections in addition to other individuals in the

population.

Figure 3: Topologies used in the Social Fabric model for

connection between individuals. (a) lBest ring topology.

(b) gBest topology.

At each time step, every individual is influenced

by one of the knowledge sources. In this simplest

version, Knowledge Sources do not know anything

about the network and the selected individuals’

position in it. The individual then transmits the name

of the influencing Knowledge Source to its

neighbours through as many hops as specified. Next,

each node counts up the number of Knowledge

source bids that it collects. It will have the direct

influence from the Knowledge Source that selected

it plus the names of the Knowledge Sources

transmitted to it by its neighbours. The Knowledge

Source that has the most votes is the winner and will

direct the individual for that time step. In case of a

tie, there are several tie breaking rules implemented

in CAT. They include, select the “most frequently

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106

used KS, “the least frequently used Knowledge

Source”, and “the Knowledge Source that selected

the individual this time”, among others. In later

sections, we will compare the performance of the

lBest and square topologies when solving an

example Engineering problem, used a benchmark.

The topographical knowledge in the belief space

is used to generate a search direction d at a given

solution x, and then use it to generate new trial

solutions in a neighbourhood of x. The

topographical knowledge structure is initialized by

sampling a solution in every cell in the grid and

creating a list of best cells. The update occurs when

a cell is divided into sub-cells when an accepted

individual’s fitness value is better than the best

solution in that cell, or if the fitness value of the

cell’s best solution has increased after a change

event is detected.

4 EXPERIMENTAL

FRAMEWORK & RESULTS

ANALYSIS

4.1 Experimental Framework

The number of individuals is fixed to 100, and the

total number of generations is 9000. If a tie is found

when the social fabric is weaved the resolution

approach used is to use the Knowledge Source that

directly affected the individual at that step.

The algorithm will be tested on a set of standard

test problems G1-G13 (Hedar and Fukushima, 2006;

Hock and Schittkowski, 1981; Koziel and

Michalewicz, 1999; Michalewicz and Schoenauer,

1996) except G2, since the objective function of

problem G2 is not differentiable. These test

problems are considered diverse enough to cover

many kinds of difficulties that constrained global

optimization faces (Hedar and Fukushima, 2006;

Wenxing Z., Ali, M., 2009), and have been used to

test performances of algorithms for constrained

global optimization.

The algorithm was used to solve each problem

30 times with 100 individuals for the population

space and a varying number of generations for each

problem depending on its complexity with a

maximum of 15000 generations for problems G1

and G15. We experimented with different kinds of

topologies through which we found that the best was

the lBest topology.

Throughout the next subsection, we will use

problem G4 for explaining how our algorithm is

used to solve such constrained optimization

problems efficiently.

4.2 Analysis of Results

In this section we report the performance of our

technique on 13 well-known test problems G1-G13.

We put in Table 1 the best known objective function

value in the second column. We report in Table 1 the

best and the worst optimal values obtained from 30

runs for each test problem. To understand quality of

the obtained solutions, we report in Table 1 for each

problem the average optimal value and the standard

deviation of the obtained objective function values

for all 30 runs. Moreover, the success rate, the

maximum number of generations before we stop

each run, used to obtain these results in 30 runs, are

reported in the third and the last columns of Table 1

for each problem respectively.

The approach used by Reynolds in (Reynolds

and Peng, 2005) did not assume that there is any

kind of connection between the individuals in the

population space. Knowledge sources will pass their

signals to the individuals at each time step. Our

approach uses different topologies to pass abstract

information obtained from the Knowledge Sources

and then weave the social fabric to allow the

individuals to pass the received info through the

assumed used topology. The amount of interaction

appears to affect the way the system solves the

presented problem of a certain complexity. Not only

the individual follows the successful Knowledge

Source but also tries to adapt through neighbours in

the built network to find a better value in the

landscape. The results in table 2 show a statistical

comparison between our new approach and some

other known approaches from literature. When

plotting the population swarms, individuals are

plotted in different shapes to indicate which

knowledge source is in control.

The population swarm plots in Figures 4-a and 4-

b show the population (individuals) moving within

the problem’s constructed landscape using the lBest

topology used by our Social Fabric (SF) approach.

Each individual is shape coded to reflect the

knowledge source that has influences it in that

generation. The best individual of a generation is

stressed using a big cross ‘X’. Since the results of

the dimensions of problems can be explained

similarly we discuss only dimensions x1 and x2.

Figure 5 shows a sample of the constructed Social

Fabric-lBest topology for problem G4.

Figures 4-a and 4-b show the initial generation

and generation 119 when running the system using

CULTURAL SWARMS - Knowledge-driven Framework for Solving Nonlinearly Constrained Global Optimization

Problems

107

the Social Fabric-lBest to illustrate how the different

knowledge sources work under the influence of the

social fabric technique to control individuals. The

Topographic Knowledge followers draw the fine-

tuning knowledge followers: Situational, Normative,

and most of the Domain Knowledge followers.

By generation 119 most of the individuals are

swarming around the best. Topographic knowledge

individuals are still exploring the space hoping to

find a better solution to report it later to the fine-

tuning knowledge followers.

(a)

(b)

Figure 4: Population swarm of dimension x1+x2 using the

lBest topology. (a) Plotted at generation 1. (b) Plotted at

generation 119.

The power behind the algorithm lies in using the

bounding boxes that the system calculates at each

time step for each of the Knowledge Sources as

illustrated in figure 6.

A bounding box represents the standard

deviation of each “dot” produced during that

generation for the mutation process. It is considered

to be the focus of the generation process by each

knowledge source. The main idea is how these

bounding boxes of the Knowledge Sources interact

(overlap area), and how focused these bounding

boxes are at each time step. The branching phase of

the algorithmic process is shown in Figures 6-a and

6-b, where initially the bounding boxes associated

with the Topographic and Normative Knowledge

Sources cover most of the space. The exploitation

process takes place with time and the bounding

boxes for the fine-grained search process have

separated from those for the coarse-grained phase

(focused search vs. wider search) and have

surrounded the optimal value for this pair of

dimensions. These bounding boxes are effectively

channelling new individuals into this area as can be

seen in figure 6-b.

Figure 5: A sample Social Fabric swarm plot for problem

G4 using lBest topology.

(a)

(b)

Figure 6: Knowledge Swarm Plot of dimension x1+x2. (a)

Plotted at generation 1. (b) Plotted at generation 119.

IJCCI 2009 - International Joint Conference on Computational Intelligence

108

Table 1: Test results for problems G1-G13.

Prob. Opt. Succ.(%) Best Av.opt. Worst S.D. #generations

G1 ‐15 100 ‐14.99993 ‐14.99986 ‐14.99984 0.000140 15000

G3 1 100 0.999987 0.999977 0.999971 0.000032 9000

G4 ‐30665.539 100 ‐30665.52 ‐30665.47 ‐30665.40 0.055110 9000

G5 5126.4981 100 5126.499 5126.501 5126.520 0.098000 1000

G6 6961.81388 100 ‐6961.81 ‐6961.779 ‐6961.550 0.088575 1000

G7 24.3062091 100 24.30590 24.30595 24.306122 0.000400 90000

G8 0.095825 100 0.095825 0.095825 0.095825 0.000000 1000

G9 680.630057 100 680.6300 680.6310 680.6315 0.015212 10000

G10 7049.250 100 7049.244 7049.247 7049.253 0.050000 1000

G11 0.75 100 0.750000 0.700001 0.750004 0.000002 1000

G12 1 100 1.000000 0.999999 0.999989 0.000018 1000

G13 0.0539498 100 0.053950 0.053953 0.053959 0.000139 15000

Table 2: Comparison of test results for problems G1-G13.

Prob.:opt.  PSO SR ASCHEA FSA Ouralg.

G1:‐15

Best

Av.

Worst

‐15.0001

‐13.2734

‐9.7012

‐15

‐14.84

N.A.

‐14.999105

‐14.993316

‐14.979977

‐14.99993

‐14.99986

‐14.99984

G3:1

Best

Av.

Worst

1.0004

0.9936

0.6674

1.000

1

0.99989

N.A.

1.0000015

0.9991874

0.9915186

0.999987

0.999977

0.999971

G4:‐30665.539

Best

Av.

Worst

‐30665.5398

‐30665.5397

‐30665.5338

‐30665.539

‐30665.5

N.A.

‐30665.5380

‐30665.4665

‐30664.6880

‐30665.52

‐30665.47

‐30665.40

G5:5126.4981

Best

Av.

Worst

5126.6467

5495.2389

6272.7423

5126.497

5128.881

5142.472

5126.5

5141.65

N.A.

5126.4981

5126.499

5126.501

5126.520

G6:6961.81388

Best

Av.

Worst

‐6961.8371

‐6961.8370

‐6961.8355

‐6961.814

‐6875.940

‐6350.262

‐6961.81

N.A.

‐6961.81388

‐6961.81

‐6961.779

‐6961.550

G7:24.3062091

Best

Av.

Worst

24.3278

24.6996

25.2962

24.307

24.374

24.642

24.3323

24.6636

N.A.

24.310571

24.3795271

24.644397

24.30590

24.30595

24.306122

G8:0.095825

Best

Av.

Worst

0.095825

0.09582

N.A.

0.095825

G9:680.630057

Best

Av.

Worst

680.6307

680.6391

680.6671

680.630

680.656

680.763

680.630

680.641

N.A.

680.63008

680.63642

680.69832

680.6300

680.6310

680.6315

G10:7049.250

Best

Av.

Worst

7090.4524

7747.6298

10533.6658

7054.316

7559.192

8835.655

7061.13

7497.434

N.A.

7059.86350

7509.32104

9398.64920

7049.244

7049.247

7049.253

G11:0.75

Best

Av.

Worst

0.7499

0.7673

0.9925

0.750

0.75

N.A.

0.7499990

0.750000

0.700001

0.750004

G12:1

Best

Av.

Worst

1.0000

1.000000

N.A.

1.000000

0.999999

0.999989

G13:0.0539498

Best

Av.

Worst

0.05941

0.81335

2.44415

0.053957

0.057006

0.216915

N.A.

0.0539498

0.2977204

0.4388511

0.053950

0.053953

0.053959

CULTURAL SWARMS - Knowledge-driven Framework for Solving Nonlinearly Constrained Global Optimization

Problems

109

5 CONCLUSIONS

The Cultural Algorithm is a stochastic

optimization method that uses evolutionary

algorithmic mechanisms to model cultural

evolution and social behaviors. Just as cultural

evolution contributes to the adaptability of human

society, CA provides an additional degree of

adaptability to evolutionary computation. In this

paper we have introduced the social fabric

influence function in the Cultural Algorithms

framework. This influence function is used to

produce population and knowledge swarms that

are used to optimally solve nonlinearly constrained

optimization problems. The SF metaphor allows

the knowledge sources to distribute their influence

through a social network. We apply this approach

to a set of well-known nonlinearly constrained

optimization problems. It turns out that the used

topology, frequency of distribution of influence,

and conflict resolution play an important role in

how efficiently the system produces knowledge

and population swarms that represent structural

patterns to solve problems.

ACKNOWLEDGEMENTS

Our thanks to Rose Ziad from Yarmouk University

Talal Ali from Wayne State University, and Imad

momani for help in developing the current design

and infrastructure of the new influence function.

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