Quantum Evolutionary Algorithm

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A Novel Quantum Dynamical Evolutionary Algorithm combined with Cultural Algorithm for Dynamical Environment

Yi Jiang, Li Yan

Journal of Convergence Information Technology, Volume6, Number 8, August 2011

A Novel Quantum Dynamical Evolutionary Algorithm combined with

Cultural Algorithm for Dynamical Environment

Yi Jiang1, 2 Li Yan3

1The School of Computer, Wuhan University

2

Wuhan University of Science and Technology3The College of Information Engineering, Wuhan University of Science and Technology

Zhong nan Branchdoi : 10.4156/jcit.vol6.issue8.42

Abstract

In this paper, we have presented a new method of the novel quantum dynamical evolutionary

algorithm combined with cultural algorithm for dynamical environment which uses belief space

representation to accommodate more types of knowledge such as History Knowledge, and

Topographical Knowledge. Also, additional influence functions have been developed to utilize these

types of knowledge. This study focuses on the knowledge needed to track an optimal solution in an

environment where the functional landscape is produced by the combination of n overlapped cones in

a 2 dimensional grid.

Keywords:Cultural Algorithm, Dynamical Evolutionary Algorithm, Dynamical Environment

1. Introduction

Research in the optimization area has focused on solving problems in static environments, while most

real-world problem environments are continuously changing (the machine may break down, jobs may be

added or removed from a schedule, or prices may change, etc.).

In the stock market, for example, people build their strategies of investment by ranking their potential

stock portfolios based on the amount of money available to invest, the waiting period before selling

(long, moderate, or short), and the risk that the investor is willing to take (high, moderate, or low). When

you add or withdraw money from your account, your stock portfolio's quality ranking may change

accordingly. This is a dynamic problem because when the investor's account balances changes, the

optimum stock combination may change as well. This is similar to the well-known dynamic problem,the Time-Varying Knapsack Problem, [Dasgupta 1992, Ghosh 1998, Goldberg 1987, Mori 1996, 1997,

1998, Ng 1995, Ryan 1997, Smith 1987], where the limit on allowable weight is changing over time

.

2. Quantum Dynamical Evolutionary Algorithm combined with Cultural

Algorithm

2.1. Cultural Algorithm

From an anthropological point of view, culture can be defined as "a system of symbolically encoded

conceptual phenomenon that is socially and historically transmitted within and between populations"

[Durham 1994]. The term "culture" was introduced for the first time by Edward Tylor in his two-volume

book "Primitive Culture" published in 1871 [Tyler, 1871]. He described culture as "that complex whole

which includes knowledge, beliefs, art, morals, customs, and any other capabilities and habits acquiredby man as a member of society." Culture's definitions assumed the existence of intellectual, generalized

knowledge within the culture's context. In human societies, culture can be viewed as a vehicle for the

storage of information that is globally accessible to all members of the society and that can be useful in

guiding their problem solving activities. In other words, cultural inheritance provides the new generation

Corresponding author.

Email addresses: [email protected](Yi Jiang)

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Yi Jiang, Li Yan


of a society with guidance and information to help individuals to adapt to their environments. Without

such information, the only way for an individual to adapt is via trial and error. DEA is the metaphorical

use of concepts, principles, and mechanisms extracted from our understanding of how natural systems

evolve to help solve complex computational problems. A model of the evolution of a cultural system is

developed to model the evolution of the cultural component of a PSO system over time as it accumulates

experience in the paper.

2.2. Quantum Dynamical Evolutionary Algorithm

2.2.1. Making qubit string

A qubit string of the length m represents a linear superposition of solutions to the problem .The

length of a qubit string is the same as the number of items. The i-th item can be selected as 1 with

probability2

| |i . Thus, a binary string of the length m is formed from the qubit string. For every bit

in the binary string, we generate a random number rfrom the range [0...1]; if2

| |ir , we set thebit of the binary string. The binary string x is determined as follows:

procedure make (x)

begini = 0

while (i < m) do

begini = i + 1

if 20,1 | |irandom

then 1ix

else 0i

x

end

end

2.2.2 Evolutionary operation

The phase of a qubit is defined with an angle as

arctan | | / | | And the product | | *| | is represented with the symbol d.

Where d stands for the quadrant of qubit phase . If dispositive, the phase lies in the first

or third quadrant, otherwise, the phase lies in the second or fourth quadrant.

Suppose that population size is n and chromosomes are represented with qubits as

1 2{ , , }nP p p p ,where ( 1,2, )j p j n is an individual of population. Quantum rotation

gate G is chose as quantum logic gate and G is

cos sin

sin cosG

Where is the rotation angle of quantum rotation gate and is defined as

* ,i ik f Where k is a coefficient and the value of k has an effect on the speed of convergence. The value of k

must be chosen reasonably. If k is too big, search grid of the algorithm is large and the solutions may

diverge or have a premature convergence to a local optimum, and if it is too little, search grid of the

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algorithm is also little and the algorithm may be in stagnant state. So k is defined as a j variable. In

CQDEA, k is defined as a variable that is relative to evolutionary generations. Thus, search grid of

CQDEA can be adjusted adaptively. For example k=10*exp (-t/maxt), where t is evolutionary

generation and maxt is a constant determined by the complexity of optimization problem.

The function ,i if determines the search direction of convergence to a global optimum.

The following lookup table (Table 1) can be used as a strategy for convergence.

Table 1. lookup table of ,i if

d1>1 d2>0 f ,i if

1 2| | | | 1 2| | | | True True +1 -1

True False -1 +1

False True -1 +1

False False +1 -1

The update procedure can be described as1

( )*t t

j j p G t p

Where t is evolutionary generation, G (t) is quantum gate and pjtis the probability amplitude of an

individual at t-th generation.

2.2.3. Dynamical Evolutionary System

For certainty of statement and not losing generality, we consider a function minimization problem

written as

min ( )x S

f x

Where f(x) is an object function and S, a feasible set of its solutions.

Suppose that a coding representation for solutions and evolutionary operators based on the code

are given. Now we take N coded solutions called particles, such as x1, x2 xN, to consist of a

dynamical system.

Correspondingly, their function values are f(x1), f(x2) f(xN). An iterative step t in the DEA

similarly like the generation in the traditional GA is called time t. Then we give the definitions of

two important quantities which control the DEA. The momentum p (t,x i) of a particle xi at time t is

defined as

( , ) ( , ) ( 1, ),i i ip t x f t x f t x Where f(t,xi) denotes the function value of the particle x i at time t. The activity a (t,x i) of a

particle xi at time t is defined as

( , ) ( 1, ) 1,i ia t x a t x If x i is selected to take part in an evolutionary operation at time t, otherwise keeps unchanging.

Incorporating these two quantities we have a criterion of selection in the DEA defined as

0

( , ) | ( , ) | ( , )t

i i i

k

slct t x p k x a t x

We can generally take a weight coefficient (0, 1) to indicate that which one of the two

terms is more significant than another in selection, namely

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0

( , ) | ( , ) | (1 ) ( , )t

i i i

k

slct t x p k x a t x

In the DEA slct(t,xi)(i=1,2,,N) are sorted in the order from small to large expressed.

The selecting strategy of DEA is that the smaller accumulation of variations of momentum and

the fewer selected times of a particle, the more possible it is selected. On the other hand, from the

procedure we notice that the slct(t,x i) must increase if xi is selected at time t.Now we derive two stopping conditions of DEA by using the momentum. The first criterion is that

if

1

| ( , ) | ,N

i

i

p t x

Where is given allowable error. From the viewpoint of statistical mechanics, that means that

all particles are sufficiently moving and dissipating energy and the dynamical system attained its

states of lowest energy.

The second criterion is that if

0

max | ( , ) | ,i

t

ix

k

p k x M

Then, the procedure will stop. Here, M is a given large number, for an example, the estimated

difference between the maximal value and the minimum value of the objective function.

2.2.4. Quantum Dynamical Evolutionary Algorithm

The structure of CQDEA is described in the following.

(1)Initialize a dynamical system(a)t=0;(b)initialize Q(t);(c)make P(t) by observing Q(t) states;(d)evaluate P(t);(e)store the best solution among P(t);(f) calculate and sort slct(t);

(2)Execute iterationrepeat

(a)t=t+1;(b)select n particles on the forefront of slct(t-1);(c)update Q(t-1) which make P(t-1) using quantum gates Q(t-1);(d)make P(t) by observing Q(t-1) states;(e)modify function values, momenta and activities of n particles;(f) save the best particle and its function value in the dynamical system;(g)modify and resort slct(t)

until{stopping criterion}

2.3. Cultural-based QDEA

The acceptance and the influence functions serve as communication channels between the belief

space and the population space. The population space influences the belief space knowledge via theacceptance function, as shown in figure 1. The influence function controls the variation operators to

guide the search. Figure 1 show how a CQDEA model is integrated into the Cultural Algorithms

framework.

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Fig 1. CQDEAs framework

The basic algorithmic steps of our version of the CQDEA are:"Generate random initial

population of p individuals. "Compute the performance score."Initialize the belief space.

Repeat.

.Select individuals with the current acceptance function.

.Update the belief space knowledge.

.Infer whether the environment has changed (by reasoning in the belief space) and update the

environment history list accordingly.

.Update the roulette wheel weight assignment for each influence function in the system.

.Generate new individuals using the DEA.

.Evaluate the new population.

Until (termination conditions are satisfied).

End

3. Experiments and Analysis

This experiment is patterned after Peter Angelines experiments in [1] Tracking Extreme Dynamic

Environments. Angeline constructed dynamic environments by starting with a simple parabolic

function in three dimensions,

f(x, y, z) = x2

+ y2

+ z2

Which has a minimum at (0, 0, 0), and then translating the function along each dimension over time.

Angeline controlled the movement of the solution using three parameters: dynamic type, specifying the

type of motion involved, step size, specifying the distance moved along each dimension, and updatefrequency, specifying the time intervals between changes in the solution. Angeline also implemented

three types of movement: linear, circular, and random. For linear motion, the function is offset step size

distance on each dimension at update frequency intervals. For circular motion the function is offset to

create a circular path with a cycle of 25 updates. For random motion, Gaussian noise is added to each

dimension with each update, resulting in movement that is random in both distance and direction.

As in Angelines experiments, step size parameters were 0.01, 0.1, and 0.5, and update frequencies

were 1, 5, and 10 iterations. A total of 270 experiments were run, reflecting each combination of

movement type, step size, and update frequency, for all five functions, implemented by both the CPSO

and the HMGA. Each experiment was repeated 100 times for 50,000 fitness function calls per run, and

the results averaged over the 100 runs.

The tables that follow indicate the average of the solutions found by each algorithm for each

combination of settings. The averages were computed from the solutions reported over the range of

10,000 to 50,000 fitness function calls. The first 10,000 fitness functions calls were omitted from thecalculation of the averages in order to allow the algorithms time to locate the moving solutions, since

the goal here it to measure the ability to track the moving solutions. For each of the five functions in

the test suite, the table of averages and one or more representative graphs are presented, as well as a

brief analysis of the performance of the algorithms. For each of the graphs, CQDEA performance

(solid line) is plotted against HMGA performance (dashed line), as measured by the error, the distance

of the best solution of the population from the actual optimal solution. The horizontal axis is the

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number of calls to the fitness function, and the vertical axis is the error. The error is plotted on a log

scale to emphasize the detail at the smaller values.

The Sphere Function is as follow

2

1

( )D

i

i

f x x

Figure 2 shows the relative performance of the CQDEA and the Hypermutation Genetic Algorithmin tracking the solution to the Sphere function. This particular graph is for continuous circular motion

(that is, the environment changes every GA generation), with a small step size (0.01 units). It is

apparent that in this circumstance CQDEA finds the solution faster and achieves better precision

throughout the life of the experiment.

As seen in Table 1 CQDEA out performs HMGA in almost every case, sometimes by an order of

magnitude in precision. There are combinations, particularly at larger step sizes with less frequent

movement, where the precision of HMGA is better than CQDEA, but the differences there are

generally small. Figure 3 shows such a situation, linear motion every ten generations with a large step

size. While overall HMGA performed about 10 percent better than CQDEA for these settings, the

graph shows that the advantage is sporadic. Taken as a whole, the performance by the two algorithms

in this case is comparable.

Fig 2. Sphere: continuous circular motion, small step size

Table 1. Sphere: average precision

Movement Frequency Step

Size

CQDEA

Average

HMGA

Averagelinear 1

5

10

0.01

0.100.500.010.10

0.500.010.100.50

0.0173

0.17520.86490.01840.1899

0.91790.21410.22621.0623

0.162

1.6788.0590.0310.347

1.6990.0160.1720.889

random 1

5

10

0.010.100.50

0.010.10

0.500.01

0.100.50

0.00920.09430.4682

0.01050.0935

0.51250.0112

0.11520.5655

0.0770.8924.163

0.0170.185

0.8610.008

0.0980.512

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Fig 3. Sphere: linear motion every 10 generations, large step size

4. Conclusion

CQDEA and HMGA appear comparable in terms of effectiveness, with CQDEA generally

producing slightly better results, although neither consistently nor significantly overall. It is also

interesting to note that in terms of physical (clock) time, HMGA takes roughly 1.5 times as long to

complete a run as does CQDEA on the same machine. This indicates that CQDEA has a lower

overhead, or fixed cost, and is thus more efficient than HMGA in performing the same tasks. Forcontinuous problems in dynamic environments, CQDEA is a viable tool for tracking optimal solutions.

5. References

[1] Gufran Ahmad, Yukio Ohsawa, Yoko Nishihara, "Cognitive Impact of Eye Movements in PictureViewing", IJIIP, Vol. 2, No. 1, pp. 1 ~ 8, 2011.

[2] S.D. Li, X. Liu, C. Cai, M.Y. Ding, "Path Planning of UAVs Using Subsection Queen-bee-assistedGenetic Algorithm", JDCTA, Vol. 5, No. 6, pp. 113 ~ 121, 2011.

[3] Sahand Khakabimamaghani, Farnaz Barzinpour, Mohammad R. Gholamian, "EnhancingEnsemble Performance through Feature Selection and Hybridization", IJIPM, Vol. 2, No. 2, pp. 78

~ 87, 2011.

[4] M. Sami Soliman, Guan-zheng Tan, "Conditional Sensor Deployment Using EvolutionaryAlgorithms", JCIT, Vol. 5, No. 2, pp. 146 ~ 154, 2010.

[5] Huafeng Wu, Chaojian Shi, Seiya Miyazaki, "Spirit: Security and Privacy in Real-TimeMonitoring System", JCIT, Vol. 5, No. 10, pp. 22 ~ 28, 2010.

[6] Toshiyuki ISHIMURA and Akinori KANASUGI, "A Design and Simulation for DynamicallyReconfigurable Systolic Array", IJIPM, Vol. 1, No. 2, pp. 18 ~ 24, 2010.

[7] Ahmed M. Mahdy and Dan Li, "A Hybrid Route Prediction Algorithm for Efficient QoS Supportin Wireless Cellular Networks", IJIPM, Vol. 1, No. 2, pp. 115 ~ 123, 2010.

[8] A. Narayanan & M. Moore, Quantum-inspired genetic algorithm, Proceedings of IEEEInternational Conference on Evolutionary Cumputation,pp.l-66, 1996.

[9] K. H. Han, J. H. Kim. Genetic quantum algorithm and its application to combinatorialoptimization problems, Proceedings of the 2000 IEEE Conferznce on Evolutionary Computation,

pp.1354-1360, 2000.

[10]T. Hey,Quantum computing: an introduction, Computing & Control Engineering Journal,pp.105-112, 1999.

[11]A. Narayanan. Quantum computing for beginners,in Proceedings of the 1999 Congress onEvolutionary Computation,pp.2231-2238, 1999.

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