A Quantum inspired Competitive Coevolution

Evolutionary Algorithm

by

Sreenivas Sremath Tirumala

under the supervision ofDr. Paul Pang and Dr. Aaron Chen

Submitted to the Department of Computer Sciencein partial fulfillment of the requirements for the degree of

Master of Computingat the

UNITEC INSTITUTE OF TECHNOLOGYOctober 2013

c⃝ Sreenivas Sremath Tirumala, MMXIII. All rightsreserved.

The author hereby grants to Unitec permission to reproduceand to distribute publicly paper and electronic copies of this

thesis document in whole or in part in any medium nowknown or hereafter created.

Abstract

Continued and rapid improvement in evolutionary algorithms has made them suit-able technologies for tackling many difficult optimization problems. Recently theintroduction of quantum inspired evolutionary computation has opened a new di-rection for further enhancing the effectiveness of these algorithms. Existing studieson quantum inspired algorithms focused primarily on evolving a single set of homo-geneous solutions. This thesis expands the scope of current research by applyingquantum computing principles, in particular the quantum superposition principle, tocompetitive coevolution algorithms (CCEA) and proposes a novel Quantum inspiredCompetitive Coevolutionary Algorithm (QCCEA). QCCEA uses a new approach toquantize candidate solution unlike previous quantum evolutionary algorithms thatuse qubit representation. The proposed QCCEA quantifies the selection procedureusing normal distribution, which empowers the algorithm to reach the optimal fitnessfaster than original CCEA. QCCEA is evaluated against CCEA on twenty bench-mark numerical optimization problems. The experimental results show that QCCEAperformed significantly better than CCEA for most benchmark functions.

Keywords - Quantum computing, evolutionary algorithm, competitive coevolution,quantum inspired, QEA, QCCEA, qubit.

NomenclatureCCEA Competitive Coevolutionary AlgorithmEA Evolutionary AlgorithmHF Hall of FameQCCEA Quantum inspired Competitive Coevolutionary AlgorithmQEA Quantum inspired Evolutionary AlgorithmQNN Quantum based Neural Network

2

Acknowledgments

Completing Master thesis is truly an achievement for me and this would not have

happened without support from countless people around me. Firstly, I would like to

express my gratitude towards my supervisor Dr.Paul Pang, for his immense knowl-

edge, understanding, support and encouragement to complete my thesis. His faith in

my potential made me more and more stronger in handling technical challenges.

A very special thanks from the bottom of my heart to Dr.Aaron Chen, who re-

defined the meaning of ’Teacher’ with his continuous knowledge, encouragement, pa-

tience and support. His vast knowledge, advises and guidance is the backbone of

my research work. He was available all the time, weekdays and weekends answering

my queries, showing his commitment and dedication towards his students. My thesis

would not have completed without him.

I would like to thank Denham Batts and Tony Gibson and other staff members

especially warehouse section of The Derek Corporation for their encouragement and

support. A Special thanks to Mark Harvey for his timely advises and lovely environ-

ment at home.

I must acknowledge my dear brother Vamsi Krishna for taking all my materialistic

responsibilities back in India. Appreciation also goes to my true brother Saravana

Kumar, for substituting me in my commitments back in India. Also thanks to my

endear sister Neeharika for proofreading.

In conclusion, I would like to thank Head of the Department of Computing

Dr.Hossein Sarrafzadeh and Programme Leader Dr.Chandimal Jayawardena for pro-

viding necessary resources and environment.

My soulful gratitude to my guru Sri Chinmoy who showed me the PATH.

3

To my mother Sarala Devi, who always believes in me

4

Contents

1 Introduction 8

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Related Work 11

2.1 Quantum inspired Evolutionary Algorithm (QEA) . . . . . . . . . . . 11

2.2 Quantum based Neural Networks (QNN) . . . . . . . . . . . . . . . . 12

2.3 Quantum inspired Artificial Bee Colony Algorithm (QABC) . . . . . 13

3 Quantum Inspired Competitive Coevolution (QCCEA) 15

3.1 Competitive Coevolution Algorithm (CCEA) . . . . . . . . . . . . . . 15

3.2 Quantum Inspired Competitive Coevolution (QCCEA) . . . . . . . . 17

4 Experiments 20

4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 Unimodal Funtions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Multimodal Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Discussion 24

6 Conclusion 29

A Test Functions f19 and f20 30

A.1 f19 Expanded Griewanks plus Rosenbrocks Function . . . . . . . . . 30

A.2 f20 Expanded Scaffers F6 Function . . . . . . . . . . . . . . . . . . . 30

5

List of Figures

2-1 qubit (QEA) Vs subsolution points (QCCEA) . . . . . . . . . . . . . 13

3-1 Overall structure of QCCEA . . . . . . . . . . . . . . . . . . . . . . . 17

5-1 Unimodal functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5-2 Basic Multimodal Functions f6 - f13 . . . . . . . . . . . . . . . . . . 26

5-3 Basic Multimodal Functions f14 - f20 . . . . . . . . . . . . . . . . . 27

6

List of Tables

4.1 BENCHMARK FUNCTIONS . . . . . . . . . . . . . . . . . . . . . . 21

4.2 CCEA Vs QCCEA ON UNIMODAL FUNCTIONS f1 − f5 . . . . . . 22

4.3 CCEA Vs QCCEA ON MULTIMODAL FUNCTIONS f6 − f20 . . . 23

7

Chapter 1

Introduction

Evolutionary algorithms (EA) have been successfully applied to various real world

problems and new techniques were developed to optimize their performance [1] [2]

[3] [4] [5]. One such attempt is application of quantum principles on evolutionary

algorithms which was conceptualized by Quantum inspired Evolutionary Algorithm

(QEA). QEA proved its efficiency in solving various types of optimization problems

[6] [7] [8] [9] [10] [11] [12]. QEA based algorithms like Quantum based Neural Network

(QNN) for optimizing neural networks was proposed in 2013 [13]. However there are

only few successful approaches to quantize evolutionary algorithms in general but none

in case of coevolutionary algorithms especially Competitive Coevolution Algorithm

(CCEA).

The ability of quantum principles to solve complex problems with high accuracy

was identified in 1950s [14], but the task of developing a mechanical computer system

(quantum computer) based on these principles was accomplished after forty years

[15]. The first quantum computer was developed by Feynman in 1981 [16] and Benioff

structured a theoretical framework for this model in 1982 [17].

In EA methods of encoding, the solution is either binary, numeric or symbolic [12]

whereas in quantum computing a solution is represented as its observations known

as state of the solution. The observations of a solution is derived using quantum

principles like uncertainty, interference, superposition and entanglement. According

to the principle of quantum superposition, when a system has multiple properties

8

and arrangements, the overall state of the system is the sum of individual states

where each state is represented as a complex number. If there are two states with

configurations a and b then the overall state is represented as c1|a⟩ + c2|b⟩ where c1

and c2 represent complex numbers.

Smallest unit of representation in classic computation is binary bit (“0” or “1”)

whereas quantum computation uses qubit which can be either “0” or “1” or super-

position of both [18]. For instance, if there are two possible states sunny or clouded

for weather, always there exist intermediate states between them depending on pa-

rameters like brightness, humidity, temperature etc.The weather may be partially

clouded which is not either sunny or clouded. However, this partially clouded state

of weather can collapse to either of the two possible states (sunny or clouded) de-

pending on brightness. Similarly when representing a qubit, the intermediate states

between 0 and 1 collapses to traditional binary bit “0” or “1” [19]. Further, the state

of a qubit is represented as amplitude that has both positive and negative values.

The qubit value depends on the shift of the amplitude in a three dimensional space

which cannot be achieved with classic computation. The detailed description of qubit

representation is presented in section II.

Algorithms based on quantum principles can be classified into two categories

namely quantum algorithms and quantum inspired or quantum based algorithms.

Quantum algorithms are explicitly designed for quantum computers. The first quan-

tum algorithm was proposed by Deutsch in 1985 called Deutschs algorithm [20] fol-

lowed by Deutch-Jozsa algorithm in 1992 [21], Shors algorithm in 1994 [22], Cleve-

Mosca in 1998 [23] and Grovers database search algorithm in 1996 [24] [25]. The

research on developing quantum inspired evolutionary algorithms by applying quan-

tum principles to classic computation algorithms was started in late 1990s and various

quantum inspired evolutionary algorithms were developed since then [26] [27] [28] [29]

[30] [31].

9

1.1 Motivation

QEA quantifies the original solution as a linear combination of two values. Since then,

all other quantum inspired algorithms followed similar approach. A novel approach

of representing the candidate solution as subsolution points of normal distribution

forms the motivation for this research.

This thesis is aimed at developing a new Quantum inspired Competitive Coevo-

lution Algorithm (QCCEA) by applying quantum principle on CCEA. Considering

the success of QEA, quantifying solutions before implementing CCEA paradigm will

result in genetic diversity with which the process of evolution becomes more efficient

resulting in vigorous competition for better solutions.

1.2 Thesis Organization

This thesis is organized as follows. A review of QEA and other QEA based algorithms

is presented in Chapter 2. Chapter 3 briefly reviews the concepts of CCEA and

proposes QCCEA in detail. Chapter 4 comprises of experimental results on CEC2013

benchmark numerical optimization problems. A performance evaluation of QCCEA

and CCEA is presented in Chapter 5. Finally Chapter 6 presents the conclusions and

direction for future work.

10

Chapter 2

Related Work

2.1 Quantum inspired Evolutionary Algorithm (QEA)

QEA was introduced by Dr. Kuk-Hyun Han which is the first ever EA based on quan-

tum computing principles [30] [31]. QEA is a population based algorithm that uses

qubit as a probabilistic representation of original solution [21]. QEA uses quantum

states to represent a candidate solution and Q-gate [32] to diversify the candidate

solution.

In QEA, qubit is the smallest unit of information which can be defined as a pair

of numbers (α, β) where |α|2 + |β|2 = 1, |α|2 gives the probability with which the

qubit will be found in “0” state, and |β|2 the probability with which the qubit will be

found in “1” state. A Qubit individual is a string of n qubits that form a candidate

solution,

⟨α1

β1

∣∣∣∣∣∣ α2

β2

∣∣∣∣∣∣ ......∣∣∣∣∣∣ αn

βn

⟩(2.1)

where 0 ≤ αi ≤ 1, 0 ≤ βi ≤ 1, |αi|2 + |βi|2 = 1, i = 1, 2, ....n and |αi|2 , |βi|2 gives

the probability with which the ith qubit will be found in state “0” and state “1”

respectively.

To further diversify the candidate solution in search process, QEA defines the

11

Q-gate as a variation operator,

U(∆θi) =

cos(∆θi) − sin(∆θi)

sin(∆θi) cos(∆θi)

(2.2)

where ∆θi, i = 1, 2, .....n , is the rotation angle of each qubit towards either 0 or 1

depending on its sign.

2.2 Quantum based Neural Networks (QNN)

Quantum based neural networks (QNN) is a QEA based approach to evolve neural

networks for network structures and weights optimization [13]. For a multilayer per-

ceptron model (MLP) neural network, the maximum number of connections cmax for

QNN is determined by

cmax = m(nh + n) +(nh + n)(nh + n− 1)

2. (2.3)

The network connectivity C is defined as ⟨α1|α2|......|αn⟩ where αi, i = 1, 2, ....., cmax

is a qubit. In this way, the network structure is quantified to have a maximum of

2Cmax candidate solutions. Similarly, the connection weights W is quantified into 2k

subspaces, W = (Qw1 , Qw2 , ...., Qwcmax) in which Qwi, i = 1, 2, ...., cmax is assumed to

contain k quantum bits or Qwi= ⟨αi,1|αi,2|.....|αcmax,k⟩. Analogous to QEA, QNN also

uses rotation gate to further diversify the candidate solution. Rotation gate is updated

according to the discrepancy between quantified solution and pre-stored best. Further

QNN employs qubit swapping as exchange operation similar to migration operation

of QEA which allows it to escape from local optima.

Typical EAs represent a candidate solution as a single point whereas QEA and

QNN quantify the candidate solution as a linear combination of two components as

shown in Fig.2-1. As the benefit, the probability of reaching the optimal solution

increases, and the convergence process to the optimal solution speeds up.

12

Figure 2-1: qubit (QEA) Vs subsolution points (QCCEA)

2.3 Quantum inspired Artificial Bee Colony Algo-

rithm (QABC)

Similar to QNN, QABC is also a QEA based algorithm developed by implementing

quantum principles on artificial bee colony paradigm [33]. In QABC, individual is

represented as a quantum vector (qubit) (similar to QEA). QABC algorithm is exe-

cuted in 5 stages as detailed below. After initializing with a set of population, each

individual qubit is projected in binary space by applying measurement operator and

fitness is calculated using

fiti =

11+fi

if fi ≥ 0

1 + abs(fi) if fi < 0(2.4)

where fi represents the quality value of considered solution. This is followed by greedy

selection where the solution with the best fitness will remain. For exploring the search

space QABC operator is used which is a quantified version of Artificial Bee Colony

13

operator which is defined as

vij

αij

βij

= Xij

αij

βij

+ ϕij

Xij

αij

βij

−Xik

αij

βij

(2.5)

where Xi, Vi are discontinued and resultant solutions and Xk is randomly selected

solution, D denotes the dimension of the problem and 1 ≤ j ≤ D, where j is the index

chosen from the dimension and ϕij is a random number within the range [−1, 1].

Population divergence is achieved by shifting qubit using quantum inheritance

operator. For a qubit A(αA, βA) the resultant diversified qubit is calculated as αB =

best(i)+L∗αA

L+1and βB =

√1− α2

B,where L is an integer coefficient and best is the best

solution to be achieved. When best(i) is guided to 1 , value of α increases with which

the probability of reaching 1 is reached and similar procedure is followed by guiding

best(i) to 0 which gives the value of β. QABC was evaluated on three numerical

optimization functions Sphere, Rosenbrock and Griewank functions against quantum

swarm algorithm and evolutionary algorithm and QABC outperforms both.

Inspired by the above quantum discipline that a candidate solution can be quan-

tified as a set of subsolution points, I address CCEA and propose QCCEA whose

candidate solutions are quantified into a non-linear combination of a fixed number of

subsolution points. Fig. 2-1 gives an illustration of the proposed Quantization using

subsolution points with a comparison to the traditional qubit quantization.

14

Chapter 3

Quantum Inspired Competitive

Coevolution (QCCEA)

3.1 Competitive Coevolution Algorithm (CCEA)

Competitive coevolution is the competitive approach of Darwins principle of survival

of the fittest where individuals compete with each other resulting in a better species

[34] [35]. In literature, three methods of competitive coevolution are prominent for

selection of the most fit individuals [36], fitness sharing, shared sampling, and Hall of

Fame (HF). In fitness sharing, every individual has a fitness sharing function which

enables grouping other individuals with the similar fitness values [37]. This helps in

identifying the most or least fit individuals in the population. Shared sampling is

implemented only for small population sizes where individuals do not compete with

the entire population, rather compete with only a sample taken from the population.

In HF, each individual competes with every other individual in the population. In the

beginning, the individuals compete with each other and the most fit individuals form a

separate list called Hall of Fame (HF). In further generations the individuals compete

with HF as well for the survival. HF gets updated as the generation progresses, thus

forming a list of fit individuals for each generation.

The algorithm 1 presents the CCEA, which uses HF strategy to evolve better

species. An individual xi is selected (step 3) among the population as the competi-

15

Algorithm 1 Competitive coevolution algorithm

1: Initialize P1, ..., PM

2: repeat3: for i = 1 to M do4: xi = Select (Pm)5: for j = 1 to M do6: yi = Select(Pm)7: if i <> j then8: X ⇐ Evaluate(xi, yj)9: end if10: end for11: Evaluate (X,HF) /*Compete with Hall of Fame*/12: Update HF13: end for14: until enough solutions are evaluated

tor for that generation. The fitness of every individual species in the population is

evaluated against xi (step 5). Resultant fit individual x is evaluated against HF. The

individuals with better fitness are added into HF for competing with further gener-

ations (Step 9). This process continues until enough solutions are evaluated. The

fitness function of the individual xi is computed as∑j

i=1(xs

i+ntime(xi)

jwhere ntime

is the number of generations since the individual is engaged in competition. New

population is generated from the evaluated individuals.

There is a continuous diversity in the results since results depends on number of

generations and time for each generation due to which the time period for reaching

required solution is not deterministic. Another interesting aspect is that, since the

selection is random, there is always a possibility of having required solution next to

worst solution. When a worst possible solution is selected by random, the evolution

process continues and time-frame for reaching the required solution cannot be guar-

anteed. However there will be significant improvement in quality of the solution with

each generation [35].

16

3.2 Quantum Inspired Competitive Coevolution (QC-

CEA)

The essence of QCCEA is that, it disseminates the candidate solution into a collection

of solution points. This is in the same discipline of QEA that enlarges the search

space and refines optimization process by quantum bit implementation [7] [13]. Here,

each solution point functions similar to that of a qubit in QEA to extend the search

capability of CCEA. From one generation to another, more genetically diverse solution

points (i.e., the normal distributions) are updated; that is the probabilities are refined

such that the overall probability of finding the global optimal solution is increased

and thus QCCEA is expected to be more resilient than CCEA towards the premature

convergence problem.

Figure 3-1: Overall structure of QCCEA

The overall structure of QCCEA is presented in Fig:3-1. A solution is selected from

17

the population and is quantized followed by fitness evaluation. This solution competes

with all other solutions in the population. The winner among these two solutions will

compete with HF resulting in the best solution b(t) for that cycle (represented in

dotted interior square in Fig:3-1). HF is updated for each generation with the b(t)

which is the best solution for that generation. The next solution is considered for

competitions and this process (shown as dotted exterior square in Fig:3-1) continues

till enough solutions are evaluated. Similar to CCEA new population is evolved with

the combination of population in HF.

Algorithm 2 QCCEA: Quantum inspired Competitive coevolution (M,N, n, b∗)

1: Initialize P1, ..., PM ; /*M solutions*/2: Initialize b(1); /*current best solutions*/3: t← 1;4: HF (1)← P (1);5: repeat6: for i = 1 to M do7: Xi = Select(Pm), 1 ≤ m ≤M ;8: Quantize Xi = {xn}Nn=1 to Xq

i = {xqn}Nn=1; /*constant variable xn is quantified as a

normal distribution vector xqn*/

9: for k = 1 to N do10: u← u+ Evaluate (xk);11: end for12: for j = 1 to M and j ̸= i do13: Yj = Select(Pm) ;14: Quantize Yj = {yn}Nn=1 to Y q

i = {yqn}Nn=1;

15: for k = 1 to N do16: v ← v+ Evaluate (yk);17: end for18: s⇐ Max(Evaluate(u), Evaluate(v));19: b(t)← max(Evaluate(s), Evaluate(HF (t))) /*Compute the best solution with current

Hall of Fame*/20: end for21: Add b(t) into HF (t);22: end for23: t← t+ 1;24: b∗ ← maxEvaluate(HF (t)); /*select the best solution from current Hall of Fame*/25: until enough solutions are evaluated

The QCCEA algorithm is detailed as Algorithm:2 and the procedure is described

as follows.

QCCEA is initialized with a population of candidate solutions P1, P2, ......., PM

(step 1) where M is the size of the population. HF and best solution of the generation

b(t) are initialized with first candidate solution P1. A solution Pm is represented as

18

Xi = {xn}Nn=1, where m,i represent an individual from the population P of size M

and N represents the width of the search space. Quantifying Xi to Xqi = {xq

n}Nn=1

where xqn is a normal distribution vector defined as xq

n = 1σ√2πe−

(x−µ)2

2σ2 . Each solution

point xqk represents the kth part of Xq

i and is evaluated which constitutes u (step

10).Similarly Yj is quantified as Y qi and evaluated at component level, thus obtaining

v (step 16).

As shown in Fig:3-1, quantified candidate solutions u and v engages in competition

(similar to CCEA) resulting in s as the solution with best fitness among u and v (step

18 of Algorithm:2). The competition between s and the HF results in b, best solution

for that generation (step 19). HF is always updated with b to maintain the most fit

solution for that generation. This process continues till the exit criteria is met. The

solution b∗ with maximum fitness among the available solutions of HF will be the

best among all candidate solutions of the population.

19

Chapter 4

Experiments

4.1 Experimental Setup

Twenty benchmark functions [38] were used in this experimental studies. All functions

are minimization problems defined as:

Minf(x), x = [x1, x2, ......., xD], (4.1)

where D is the dimensionality of x. According to [39] [40] [41] , twenty benchmark

functions is a sufficient number of functions to find out whether the proposed QCCEA

is better (or worse) than CCEA, and why. More multimodal functions are used since

the local minima increases exponentially with the dimension [42] [43] resulting in

increase of complexity which is an ideal challenge for many optimization algorithm

evaluations.

The mathematical description of the used functions is listed in Table 4.1, where

functions f1-f5 are unimodal functions and f6-f20 are basic multimodal functions.The

detailed description of test functions f19 and f20 is given in the Appendix A. In the

table 4.1, M1,M2, ......,M10 are orthogonal (rotation) matrices generated from stan-

dard normally distributed entries by Gram-Schmidt ortho normalization which is a

common practice for numerical optimization and o = [o1, o2, ......., oD] are the shifted

global optimums which are randomly distributed in [−80, 80]D. All test functions

20

are shifted to o and scalable. Λα is a diagonal matrix in D dimensions with the ith

diagonal element as λii = αi−1

2(D−1) , i = 1, 2, ......, D.

T βasy : if xi > 0, xi = x

1+β i−1D−1

√xi

i , for i = 1, ....., D

Tosz : for xi = sign(xi)exp(x̄i + 0.049(sin(c1x̄i) + sin(c2x̄i))), for i = 1 and D

where x̄i =

log(|xi|) if xi ̸= 0

0 otherwise, sign(xi) =

−1 if xi < 0

0 if xi = 0

1 otherwise

c1 =

10 if xi > 0

5.5 otherwise, and c2 =

7.9 if xi > 0

3.1 otherwise

Table 4.1: BENCHMARK FUNCTIONS

Test function Parameter z fmin

f1(x) =∑D

i=1 z2i + f∗

1 o -1400

f2(x) =∑D

i=1(106)

i−1D−1 z2i + f∗

2 Tosz(M1(x− o)) -1300

f3(x) = z2i + 106∑D

i=2 z2i + f∗

3 M2T 0.5asy(M1(x− o)) -1200

f4(x) = 106z2i +∑D

i=2 z2i + f∗

4 Tosz(M1(x− o)) -1100

f5(x) =

√∑Di=1 |zi|

2+ i−1D−1 + f∗

5 x− o -1000

f6(x) =∑D−1

i=1 (100(z2i − zi+1)2 + (zi − 1)2) + f∗

6 M1(2.048(x−o)

100) + 1 -900

f7(x) = ( 1D−1

∑D−1i=1 (

√zi +

√zi sin

2(50z0.2i )))2 + f∗7

√y2i + y2i+1 for i = 1, ....., D,

y = Λ10M2T 0.5asy(M1(x− o))

-800

f8(x) = −20exp(−0.2√

1D

∑Di=1 z

2i )

−exp( 1D)∑D

i=1 cos(2Πzi)) + 20 + e+ f∗8

Λ10M2T 0.5asy(M1(x− o)) -700

f9(x) =∑D

i=1(∑kmax

k=0 [ak cos(2πbk(zi + 0.5))])

−D∑kmax

k=0 [ak cos(2πbk.0.5)] + f∗9

Λ10M2T 0.5asy(M1

0.5(x−o)100

)a = 0.5, b = 3, kmax = 20

-600

f10(x) =∑D

i=1z2i

4000−

∏Di=1 cos(

zi√i) + 1 + f∗

10 Λ100M1600(x−o)

100) -500

f11(x) =∑D

i=1(z2i − 10 cos(2πzi) + 10) + f∗

11 Λ10T 0.2asy(Tosz(

5.12(x−o)100

)) -400

f12(x) =∑D

i=1(z2i − 10 cos(2πzi) + 10) + f∗

12 M1Λ10M2T 0.2asy(Tosz(

5.12(x−o)100

)) -300

f13(x) =∑D

i=1(z2i − 10 cos(2πzi) + 10) + f∗

13 M1Λ10M2T 0.2asy(Tosz(y)) -200

f14(z) = 418.9829×D − σDi=1g(zi) + f∗

14 Λ10(1000(x−o)

100) + 4.209687462275036e+ 002 -100

f15(z) = 418.9829×D − σDi=1g(zi) + f∗

15 Λ10M1(1000(x−o)

100) + 4.209687462275036e+ 002 100

f16(x) =10D2

∏Di=1(1 + i

∑32j=1

|2f zi−round(2f zi)|2f

)D101.2 M2Λ100(M1

5(x−o)100

) 200

f17(x) = min(∑D

i=1(x̄i

−µ0)2, dD + s∑D

i=1(x̄i − µ1)2)

+10(D −∑D

i=1 cos(2πz̄i)) + f∗17

Λ100(x̄− µ0)

y =10(x−o)

100, x̄i = 2sign(x∗

i )yi+µ0, for i = 1, 2, ...., D

300

f18(x) = min(∑D

i=1(x̄i − µ0)2,

dD + s∑D

i=1(x̄i − µ1)2)

+10(D −∑D

i=1 cos(2πz̄i)) + f∗18

M2Λ100(M1(x̄− µ0)),

y =10(x−o)

100, x̄i = 2sign(y∗i )yi

+µ0, for i = 1, 2, ...., D

400

f19(x) = g1(g2(z1, z2)) + g1(g2(z2, z3)) + .....+g1(g2(zD−1, zD)) + g1(g2(zD, z1)) + f∗

19M1(

5(x−o)100

) + 1 500

f20(X) = g(z1, z2) + g(z2, z3) + ......+g(zD−1, zD) + g(zD, z1) + f∗

20M2T 0.5

asy(M1(x− o)) 600

For all experiments, same population size (100), same search range and same

dimension is used. Same initial population is maintained for both CCEA and QCCEA.

For simplicity, all the test functions are set with the same search range as [−100, 100]D.

The Dimension D is set as 10 for all the functions. The experiment is performed for

21

3000 generations and each algorithm is executed 25 times.

4.2 Unimodal Funtions

The first set of experiments are on functions f1 − f5. The obtained average results

of 25 runs evolutions are presented in Table 4.2.

Table 4.2: CCEA Vs QCCEA ON UNIMODAL FUNCTIONS f1 − f5

No. Functions Benchmark CCEA QCCEA T-Test Outcomesf1 Sphere -1400 -1241.50 -1327.20 5.82197 x 10−55

f2 Rotated High Conditioned Elliptic -1300 -1274.60 -1215.20 5.79889 x 10−49

f3 Rotated Bent Cigar -1200 -941.60 -1135.50 8.23198 x 10−75

f4 Rotated Discus -1100 -1054.8 -1082.6 1.15199 x 10−31

f5 Different Powers -1000 -903.9 923.6 2.76096 x 10−31

4.3 Multimodal Functions

The second set of experiments are aimed at 15 basic multimodal functions f6-f20.

Multimodal functions are often regarded difficult to be optimized because of their

multiple local minimum values exists in a huge local optima search space [39]. The

global optima of functions f11-f15 is far from their local optima, whereas function

f6 has a narrow valley between its local and global optima. Functions f16 and f18

are extremely difficult to be optimized since their local minimum values are present

everywhere inside the search space and are non differentiable.

Table 4.3 summarizes the average fitness values obtained when applying proposed

QCCEA to optimize the considered functions. It contains also the average fitness val-

ues of CCEA for comparison. Fig.5-2 and Fig. 5-3 shows the progressive convergence

of all multimodal functions.

22

Table 4.3: CCEA Vs QCCEA ON MULTIMODAL FUNCTIONS f6 − f20

No. Functions Benchmark CCEA QCCEA T-Test Outcomesf6 Rotated Rosenbrock’s -900 -741.0 -772.8 1.03462 x 10−34

f7 Rotated Schaffer’s F7 -800 -688.7 -762.6 8.83422 x 10−51

f8 Rotated Ackley’s -700 -592.0 -646.9 1.99006 x 10−45

f9 Rotated Weierstrass -600 -490.5 -560.8 4.87262 x 10−54

f10 Rotated Griewank’s -500 -357.2 -474.4 1.84575 x 10−63

f11 Rastrigin’s -400 -332.9 -336.4 2.40845 x 10−18

f12 Rotated Rastrigin’s -300 -177.5 -224.3 2.15288 x 10−45

f13 Non-Continuous Rotated Rastrigin’s -200 -79.3 -97.3 1.7107 x 10−25

f14 Schwefel’s -100 -92.4 -88.7 0.002737591f15 Rotated Schwefel’s 100 84.0 72.1 4.89885 x 10−19

f16 Rotated Katsuura 200 165.0 183.5 3.0037 x 10−27

f17 Lunacek Bi Rastrigin 300 222.6 253.4 1.15312 x 10−38

f18 Rotated Lunacek Bi Rastrigin 400 290.9 309.1 4.1089 x 10−22

f19 Expanded Griewank’s plus Rosenbrock’s 500 289.3 263.9 1.18431 x 10−34

f20 Expanded Scaffer’s F6 600 412.5 430.3 9.67914 x 10−20

23

Chapter 5

Discussion

Fig.5-1 shows respectively the progress of average fitness of CCEA and QCCEA for

the five test functions over 25 runs. As seen from the results, QCCEA performs

consistently closer to the optimum or near optimum than CCEA for four of total five

unimodal functions. For function f2, QCCEA lags behind CCEA with a statistically

negligible difference of 4.8%.

Sphere Function f1 is a commonly used initial test function for performance eval-

uation of numerical optimization algorithms. For function f1 both CCEA and QC-

CEA exhibited similar performance at the beginning (till 721 generations) but at the

later stages, QCCEA has improved progressively due to quantization. QCCEA has

dominated in 2279 generations out of 3000 for function f1.

The biggest performance superiority of QCCEA occurs with Rotated Bent Cigar

function f3 which is 17%. QCCEA achieved better convergence rate than CCEA due

to its extensive search ability for the same number of generations while CCEA was

caught within a small search space.

As shown in Table 4.3, QCCEA outperforms CCEA for twelve out of fifteen func-

tions. Moreover, QCCEA performed equally well for two out of three CCEA domi-

nated functions. The highest performance dominance of QCCEA over CCEA is 24%

, which occurs with f10, the Rotated Griewank’s function as shown in Fig. 5-2 .

For Rotated Ackley’s function f8 shown in Fig.5-2, QCCEA reaches the best

fitness (at generation 1500) 500 generations faster than CCEA. Interestingly, both

24

f1 f2

f3 f4

f5

Figure 5-1: Unimodal functions

25

f6 f7

f8 f9

f10 f11

f12 f13

Figure 5-2: Basic Multimodal Functions f6 - f13

26

f14 f15

f16 f17

f18 f19

f20

Figure 5-3: Basic Multimodal Functions f14 - f20

27

QCCEA and CCEA reaches the same fitness value at generation 2000, but QCCEA

wins over CCEA for rest of the generations. Both the algorithms are unable to retain

their search boundaries to the near optimal value since function f8 is an asymmetrical

function.

For Rotated Rastrigin’s function f12, both algorithms could not even reach near

optimal value as shown in Fig.5-2. Nevertheless, QCCEA’s performance is outstand-

ing. In this, QCCEA reaches 25% away from the optima, which is 20% closer to the

optima than the CCEA.

For Lunacek Bi Rastrigin function f17, both QCCEA and CCEA starts at the

same point, but the performance of CCEA declines within the first 500 generations

and maintains the same convergence rate for the remaining 2500 generations as ob-

served in Fig.5-3. In contrast, QCCEA forms average 12.1% superiority to CCEA

during the first 500 generations, and this superiority is retained until the last gener-

ation.

From the above analysis, it is evident that QCCEA has a controlled performance

irrespective of shape, properties and local minimum implications of the functions

whereas CCEA was unable to adjust itself accordingly. For rest of the QCCEA domi-

nated functions, the progressive convergence of QCCEA against CCEA is consistent.

For the three functions f14,f15 and f19 where QCCEA fall short of CCEA, the

difference in the average fitness is 1%, 10.6% and 7.5% respectively, as represented in

Fig.5-3

28

Chapter 6

Conclusion

This thesis presented QCCEA, a new quantum inspired competitive coevolution al-

gorithm. In QCCEA, candidate solution is represented through a combination of a

set of solution points which are jointly described through normal distributions. This

is different from traditional quantization methods such as QEA [7] and QNN [13].

The performance of the proposed QCCEA is evaluated on 20 benchmark numerical

optimization functions published in CEC 2013 [38]. According to the obtained exper-

iment results, QCCEA is outperforming CCEA over 16 out of 20 test functions, and

the search speed of the QCCEA is noticeably faster in that QCCEA reaches global

optima or near optima in less number of generations than that of CCEA. Statistical

study based on the obtained results also shows that QCCEA can maintain a good

balance between exploitation and exploration over the whole search space. It is there-

fore more effective in identifying global optimal solutions. Based on these findings, it

can be finally concluded that quantum computing principles developed in this paper

can help to improve the performance of coevolutionary algorithms.

QCCEA, as presented in this paper, is only suitable for tackling numerical opti-

mization problems. Looking into the future, it remains interesting to see how QCCEA

can be further extended in order to solve combinatorial optimization problems that

are of particular interests in practical applications. It is also interesting to extensively

evaluate the potential and effectiveness of QCCEA for general-purpose machine learn-

ing and decision making tasks.

29

Appendix A

Test Functions f19 and f20

A.1 f19 Expanded Griewanks plus Rosenbrocks Func-

tion

Griewank’s Function

g1(x) =∑D

i=1x2i

4000−∏D

i=1 cos(xi√i) + 1

Rosenbrock’s Function

g2(x) =∑D−1

i=1 (100(x2i − xi+1)

2 + (xi − 1)2)

f19(x) = g1(g2(z1, z2)) + g1(g2(z2, z3)) + .....+ g1(g2(zD−1, zD)) + g1(g2(zD, z1)) + f ∗19

z = M1(5(x−o)100

) + 1

A.2 f20 Expanded Scaffers F6 Function

g(x, y) = 0.5 +(sin2(√

x2+y2)−0.5)

(1+0.001(x2+y2)),

f20(X) = g(z1, z2) + g(z2, z3) + ......+ g(zD−1, zD) + g(zD, z1) + f ∗20

z = M2T0.5asy(M1(x− o))

30

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