Recent Advances in Differential Evolution Yong Wang Lecturer, Ph.D. School of Information Science...

Recent Advances in Differential Evolution

Yong Wang Lecturer, Ph.D.

School of Information Science and Engineering,Central South University

[email protected]

2

Outline of My Talk

Introduction to Differential Evolution

The State-of-the-Art of Differential Evolution

Composite Differential Evolution

Orthogonal Crossover based Differential Evolution

Conclusion

3

Evolutionary Algorithms

• What are evolutionary algorithms (EAs)? EAs are intelligent optimization and search techniques inspi

red by nature

• Why evolutionary algorithms (EAs)?

• The framework of evolutionary algorithms (EAs)

Is it the optimal solution?

The optimal solution!

x

f(x)

The optimal solution!

Population

Parent Set

Selection

the first individualthe second individual

the NPth individual

New Solutions

CrossoverMutation

Replacement

xy

f(x,y)

020

4060

0

20

40

60-10

-5

0

5

10

4

Differential Evolution (1/2)

• Differential evolution (DE), proposed by Storn and Price in 1995, is one of the main branches of evolutionary algorithms (EAs).

• DE includes three main operators, i.e., mutation operator, crossover operator, and selection operator.

• Currently, DE has been successfully used in various fields.

5

Differential Evolution (2/2)

• The algorithmic framework of DE

Remark: mutation + crossover = trial vector generation strategy

the target vectors

6

The Mutation Operators

• rand/1 )( ,3,2,1, GrGrGrGi xxFxv

• rand/2 )()( ,5,4,3,2,1, GrGrGrGrGrGi xxFxxFxv

)( ,2,1,, GrGrGbestGi xxFxv

• best/1

)()( ,2,1,,,, GrGrGiGbestGiGi xxFxxFxv

)()( ,4,3,2,1,, GrGrGrGrGbestGi xxFxxFxv

• best/2

• current-to-best/1

• current-to-rand/1 )()( ,3,2,,1,, GrGrGiGrGiGi xxFxxrandxv

Remark: r1, r2, r3, r4, and r5 are different indexes uniformly randomly selected from , is the best individuals in the current population.}{\},,1{ iNP

the base vector

the difference vector

Gbestx ,

the fashion the base vector has been selected

the number of the difference vector

the scaling factor

the scaling factor

The scaling factor F plays a very important role in mutation.

7

The Characteristics of the Mutation Operators (1/3)

• rand/1 )( ,3,2,1, GrGrGrGi xxFxv

• Characteristics rand/1 is the most commonly used mutation operator in the

literature. All vectors for mutation are selected from the population at

random and, consequently, it has no bias to any special search directions and chooses new search directions in a random manner.

It usually demonstrates slow convergence speed and bears stronger exploration capability.

8


• rand/2 )()( ,5,4,3,2,1, GrGrGrGrGrGi xxFxxFxv

• Characteristics In rand/2, two difference vectors are added to the base

vector, which might lead to better perturbation than the strategies with only one difference vector.

It can generate more different trial vectors than the rand/1 mutation operator with respect to the same population.

When using rand/2, the diversity of the population can be kept, however, it has a side effect on the convergence speed of DE.

9


)( ,2,1,, GrGrGbestGi xxFxv

• best/1

)()( ,2,1,,,, GrGrGiGbestGiGi xxFxxFxv

)()( ,4,3,2,1,, GrGrGrGrGbestGi xxFxxFxv

• best/2

• current-to-best/1

• Characteristics best/1, best/2 and current-to-best/1 usually have the fast converge

nce speed and perform well when solving unimodal problems. They are easier to get stuck at a local optimum and thereby lead to

a premature convergence when solving multimodal problems. The best/1 is a degenerated case of the current-to-best/1 with the fi

rst scaling factor F being equal to 1.

10

The Crossover Operators (1/2)

• Binomial crossover

mix

, ,1, ,2, , , , ,( , , , , , )i G i G i G i r G i D Gx x x x x

, ,1, ,2, , , , ,( , , , , , )i G i G i G i r G i D Gu u u u u

, ,1, ,2, , , , ,( , , , , , )i G i G i G i r G i D Gv v v v v

the mutant vector

the trial vector

the target vector

rand1≤CR

rand1>CR rand2>CR

rand2≤CR

Giu ,

Gix ,

is always different from

11

mix

The Crossover Operators (2/2)

• Exponential crossover

, ,1, , , , , , ,( , , , , , , )i G i G i L G i L r G i D Gx x x x x

, ,1, , , , , , ,( , , , , , , )i G i G i L G i L r G i D Gu u u u u

the trial vector

the mutant vector

the target vector

Pr(r≥v)= CRv-1

, ,1, , , , , , ,( , , , , , , )i G i G i L G i L r G i D Gv v v v v

The crossover control parameter CR plays a very important role in crossover.

12

The Characteristics of the Crossover Operators

• Characteristics Binomial crossover is similar to discrete crossover in

genetic algorithm. Exponential crossover is functionally equivalent to two-point

crossover in genetic algorithm. Exponential crossover has the capability in maintaining the

linkage among variables and the building block. Binomial crossover may destroy building block.

13

DE Variations

• By combining different mutation operators and different crossover operators, we can obtain different DE variants.

• DE/x/y/z DE: differential evolution x: the fashion the base vector has been selected y: the number of the difference vector z: the type of the crossover operator; “bin” represents the bi

nomial crossover and “exp” represents the exponential crossover

• DE/rand/1/bin, DE/rand/1/exp, DE/rand/2/bin, …

14

The Illustrative Graph of DE/rand/1/bin

the triangle denotes the trial vector Giu ,

Gix ,

Grx ,2

Grx ,1

Giv ,

0

Grx ,3

1x

2x

)( ,3,2 GrGr xxF

base vector

perturbed vectors

15

On Rotation Invariance (1/3)

• Why the rotation invariance is very important for optimization algorithms We have no a prior knowledge about the topology structure

of the optimization problems

-5

0

5

-5

0

50

20

40

60

80

16


• In DE, the crossover control parameter CR controls the rotation invariance to a certain degree

CR=0.0 CR=0.5 CR=1.0

S. Das, and P. N. Suganthan. Differential evolution: A survey of the state-of-the-art. IEEE Transactions on Evolutionary Computation, vol. 15, no. 1, pp. 4-31, 2011.

17


• current-to-rand/1 is a rotation-invariant strategy

)( ,3,2,1, GrGrGrGi xxFxv

)()( ,3,2,,1,, GrGrGiGrGiGi xxFxxrandxv

rand/1

)( ,,,, GiGiGiGi xvrandxu

)()( ,3,2,,1,, GrGrGiGrGiGi xxFrandxxrandxu

arithmetic crossover

Remark: current-to-rand/1 can be considered as rand/1 + arithmetic crossover, in which the crossover control parameter CR is unnecessary

arithmetic crossover

binomial crossover/

exponential crossover

18

Outline of My Talk





Conclusion

19

The Current Research Directions of DE

• The DE performance mainly depends on two components trial vector generation strategy (i.e., the mutation and

crossover operators) control parameters (i.e., the population size NP, the scaling

factor F, and the crossover control parameter CR).

• Much effort has been made to improve the performance of DE Introduction of new trial vector generation strategy for

generating new solutions Tuning the control parameters (static/deterministic,

dynamic/adaptive, and self-adaptive) Hybridization of DE with other operators or methods Use of multiple populations (distributed DE)

20

Six Representative DE

• jDE (self-adaptive parameters in DE, IEEE TEC, 2006, 10(6))

• DEahcSPX (DE with adaptive hill-climbing and simplex crossover, IEEE T

EC, 2008, 12(1))

• SaDE (DE with strategy adaptation, IEEE TEC, 2009, 13(2))

• JADE (adaptive DE with optional external archive IEEE TEC, 2009, 13(5))

• DEGL (DE using a neighborhood-based mutation operator, IEEE TEC, 20

09, 13(3))

• ODE (opposition-based DE, IEEE TEC, 2008, 12(1))

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jDE

• Main motivation How to self-adaptively adjust the scaling factor F and the

crossover control parameters CR of DE

• Main idea F and CR are applied at individual level

… … …

2,Gx1,Gx

,NP Gx

1,GF 1,GCR

2,GF2,GCR

,NP GF ,NP GCR

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DEahcSPX (1/2)

• Main motivation Incorporating local search (LS) heuristics is often very

useful in designing an effective evolutionary algorithm for global optimization.

• Main challenges of XLS the length of the XLS the selection of individuals which undergo the XLS the choice of the other parents which participate in the

crossover operation whether deterministic or stochastic application of XLS

should be used

Crossover-based LS (XLS)

Local improvement process (LIP) oriented LS

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DEahcSPX (2/2)

• Main techniques At each generation, firstly, the best individual with other np in

dividuals randomly chosen from the population are selected to participate in the simplex crossover (SPX).

One offspring is produced and if the offspring is better than the best individual, then it will be used to replace the best individual.

Afterward, DE is implemented.

adaptive hill-climbing (ahc)

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SaDE (1/4)

• Main motivation At different stages of evolution, different trial vector

generation strategies coupled with different control parameter settings may be required in order to achieve the best performance.

• Main idea Adaptively adjust the trial vector generation strategies and the

control parameters simultaneously by learning from their previous experiences.

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SaDE (2/4)

• How to adapt the trial vector generation strategy Use four trial vector generation strategies to construct the

strategy candidate pool

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SaDE (3/4)

• How to adapt the trial vector generation strategy For each trial vector generation strategy at generation G, Sa

DE records: nk,G : the number of the trial vectors generated by the kth strate

gy nsk,G: the number of the trial vectors generated by the kth strate

gy which can enter the next generation

During the first LP generations, each trial vector generation strategy is chose with the same probability. When the generation number G is larger than LP, the probability, pk,G, of using control parameter setting k is calculated as follows:

ςn

nsS G

LPGg gk

G

LPGg gk

Gk +=∑∑

1-

-= ,

1-

-= ,

, ∑4

1= ,

,, =

k Gk

GkGk

S

Spand

avoid all the success rates being equal to zero

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SaDE (4/4)

• How to adapt F and CR the parameter F is approximated by a normal distribution with mea

n value 0.5 and standard deviation 0.3, denoted by N(0.5,0.3). CR obeys a normal distribution with mean value CRm and standard

deviation Std=0.1, denoted by N(CRm,Std) where CRm is initialized as 0.5.

CRMemoryk is used to store those values with respect to the kth strategy that have generated trial vectors successfully entering the next generation within the previous LP generations.

During the first LP generations, CR values with respect to kth strategy are generated by N(0.5,0.1).

At each generation after LP generations, the median value stored in CRMemoryk will be calculated to overwrite CRmk. Then, CR values can be generated according to N(CRmk,0.1) when applying the kth strategy.

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How to exploit the advantages and overcome the disadvantages of the current-to-best/1 and how to adapt F and CR during the evolution

JADE (1/4)

• Main motivation The current-to-best/1 benefit from its fast convergence by

incorporating best solution information in the evolutionary search. However, the best solution information may also cause problems such as premature convergence due to the resultant reduced population diversity.

A well-designed parameter adaptation scheme is usually beneficial to enhance the robustness of an algorithm.

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JADE (2/4)

• A new mutation operator: current-to-pbest/1

• The characteristics of current-to-pbest/1 Any of the top 100p% solutions can be randomly chosen

to play the role of the single best solution in DE/current-to-best.

Recently explored inferior solutions, when compared to the current population, provide additional information about the promising progress direction. Denote A as the set of archived inferior solutions and P as the current population.

is randomly chosen from the union .

)()( ,2,1,,,, GrGriGip

GbestiGiGi xxFxxFxv

Grx ,2

AP

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JADE (3/4)

• How to adapt CR

( ,0.1)i i CRCR randn

The mean is initialized to be 0.5 and then updated at the end of each generation as:

where c is a positive constant between 0 and 1, SCR is the set of a

ll successful crossover probabilities CRi at generation G. and meanA

(·) is the usual arithmetic mean.

(1 ) ( )CR CR A CRc c mean S

CR

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JADE (4/4)

• How to adapt F

( ,0.1)i i FF randc

The location parameter of the Cauchy distribution is initialized to be 0.5 and then updated at the end of each generation as

where SF is the set of all successful mutation factors in generation G

and meanL(·) is the Lehmer mean

(1 ) ( )F F L Fc c mean S

2

( ) F

F

F SL F

F S

Fmean S

F

F

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How to balance the exploration and exploitation in the current-to-best/1

DEGL (1/3)

• Main motivation A proper tradeoff between exploration and exploitation is

necessary for the efficiency and effectiveness of a population-based stochastic search method.

The current-to-best/1 of DE favors exploitation only, since all the vectors are attracted by the same best position found so far by the entire population.

As a result of such exploitative tendency, in many cases, the population of DE may lose its global exploration abilities within a relatively small number of generations.

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DEGL (2/3)

• Main idea

Remark: w controls the balance between the exploration and exploitation

global mutation model

local neighborhood model

how to define the neighborhood

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DEGL (3/3)

• How to set the parameter w increasing weight factor

linear increment exponential increment

random weight factor self-adaptive weight factor

the weight factor associated with the best individual of the population

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ODE (1/3)

• Main motivation All population-based optimization algorithms, no exception

for DE, suffer from long computational times because of their evolutionary/stochastic nature.

In the absence of a priori information about the solution, we usually start with random guesses. The computation time, is related to the distance of these initial guesses from the optimal solution.

• Main idea By using the current solution and its opposite solution, the

convergence speed of DE can be enhanced (opposition-based learning).

a b

the optimal solutionthe current solution the opposite solution

(a+b)/2

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ODE (2/3)

• Basic definitions of opposition-based learning Opposite number: Let be a real number. The

opposite number is defined as . Opposite solution: Let be a solution in D-

dimensional space, where and . The opposite point is completely defined by its components

. Opposition-Based Comparison: Let be a

solution in D-dimensional space and its objective function value. According to the definition of the opposite solution, is the opposite solution of . If , then can be replaced with .

[ , ]x a bx x a b x

1( , , )Dx x x

1, , Dx x [ , ]i i ix a b

i i i ix a b x

1( , , )Dx x x

( )f x

1( , , )Dx x x

( ) ( )f x f x 1( , , )Dx x x

x

x

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ODE (3/3)

• Opposition-based population initialization

• Opposition-based generation jumping

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Outline of My Talk





Conclusion

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Composite Differential Evolution (CoDE)

• Motivation During the last decade, DE researchers have suggested

many empirical guidelines for choosing trial vector generation strategies and control parameter settings.

some trial vector generation strategies are suitable for the global search and some others are useful for rotated problems

some control parameter settings can speed up the convergence and some other settings are effective for separable functions

However, these experiences have not yet systematically exploited in DE algorithm design.

whether the performance of DE can be improved by combining several effective trial vector generation strategies with some

suitable control parameter settings

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• Main idea

strategy candidate pool parameter candidate pool

Y. Wang, Z. Cai, and Q. Zhang, “Differential evolution with composite trial vector generation strategies and control parameters.” IEEE Transactions on Evolutionary Computation, vol. 15, no. 1, pp. 55-66, 2011.

DE/rand/1/bin

DE/rand/2/bin

DE/current-to-rand/1

F=1.0, CR=0.1

F=0.8, CR=0.2

F=1.0, CR=0.9

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• In general, we expect that the chosen trial vector generation strategies and control parameter settings show distinct advantages.

• Thus, they can be effectively combined to solve different kinds of problems.

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• Basic properties of the strategy candidate pool DE/rand/1/bin has stronger global exploration ability, and it

is effective when solving multimodal problems. DE/rand/2/bin may lead to better permutation than

DE/rand/1/bin, since the former uses two difference vectors. DE/current-to-rand/1 is rotation-invariant and suitable for

rotated problems.

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• Basic properties of the parameter candidate pool A large value of F can make the mutant vectors distribute widely in

the search space and can increase the population diversity. A low value of F makes the search focus on neighborhoods of the

current solutions, and thus it can speed up the convergence. A large value of CR can make the trial vector very different from the

target vector. Therefore, the diversity of the offspring population can be encouraged.

A small value of CR is very suitable for separable problems, since in this case the trial vector may be different from the target vector by only one parameter and, as a result, each parameter is optimized independently.

44


• Basic properties of the parameter candidate pool When combined with the three strategies, [F=1.0,CR=0.1] is

for dealing with separable problems. [F=1.0,CR=0.9] is mainly to maintain the population diversity

and to make the three strategies powerful in global exploration.

[F=0.8,CR=0.2] encourages the exploitation of the three strategies in the search space and thus accelerates the convergence speed of the population.

Conclusion: the selected strategies and parameter settings exhibit distinct advantages and, therefore, they can complement one another for solving optimization problems of different characteristics.

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• The main framework

target vector

the first trial vector

the third trial vector

the second trial vector the best trial vector

combining each trial vector generation strategies with one control parameter

setting randomly selected

combining each trial vector generation strategies with one control parameter

setting randomly selected

,i Gx

comparison

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• The experimental results 25 test functions proposed in the IEEE CEC2005 were used t

o study the performance of the proposed CoDE unimodal functions F1–F5

basic multimodal functions F6–F12

expanded multimodal functions F13–F14

hybrid composition functions F15–F25

The average and standard deviation of the function error value were recorded for measuring the performance of each algorithm

For each test function, 25 independent runs were conducted with 300,000 function evaluations (FES) as the termination criterion

Wilcoxon’s rank sum test at a 0.05 significance level was conducted on the experimental results

*| ( ) ( ) |bestf x f x

the best solution found by the algorithm in a run

the global optimum of the test function

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• The experimental results Comparison with four state-of-the-art DE

“ －” , “ ＋” , and “≈” denote that the performance of the corresponding algorithm is worse than, better than, and similar to that of CoDE, respectively.

Overall, CoDE is better than the four competitors.

basic multimodal functions expanded multimodal functionshybrid composition functions

CoDE is the best

unimodal functions CoDE is the second best

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• The experimental results Comparison with CLPSO, CMA-ES, and GL-25

“ －” , “ ＋” , and “≈” denote that the performance of the corresponding algorithm is worse than, better than, and similar to that of CoDE, respectively.

Overall, CoDE significantly outperforms CLPSO, CMA-ES, and GL-25.

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• The experimental results Random selection of the control parameter settings (CoDE) Adaptive selection of the control parameter settings (adaptiv

e CoDE) Adaptive CoDE VS CoDE

the adaptive CoDE outperforms CoDE on one unimodal functionCoDE wins the adaptive CoDE on another unimodal functionCoDE wins on two hybrid composition functions

Overall, CoDE is slightly better than the adaptive CoDE.

50

Outline of My Talk





Conclusion

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• Motivation The crossover operators of DE can only generate a vertex

of a hyper-rectangle defined by the mutant and target vectors.

Therefore, the search ability of DE may be limited.

the triangle denotes the trial vector

Gix ,

Grx ,2

Grx ,1

Giv ,

0

Grx ,3

1x

2x

)( ,3,2 GrGr xxF

base vector

perturbed vectors

whether the search ability of DE can be enhanced by effectively probing the hyper-rectangle defined by the

mutant and target vectors

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• Orthogonal crossover (OX) can make a systematic and rational search in a region defined by the parent solutions.

• OX is based on orthogonal design.

• Orthogonal design: an example

Factors (K)

the temperature the amount of fertilizer the pH value of the soil

Levels (Q)

20 ℃ 100 g/m2 6

25 ℃ 150 g/m2 7

30 ℃ 200 g/m2 8

The number of all the experiments is QK

the main aim of orthogonal design is to choose several representative combinations

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• How to implement the orthogonal design Orthogonal array: An orthogonal array for K factors with Q le

vels and M combinations is often denoted by LM(QK).

The orthogonality of an orthogonal array means that: each level of the factor occurs the same number of times in ea

ch column each possible level combination of any two given factors occur

s the same number of times in the array.

a level combination

a factor

a level

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• Quantization orthogonal crossover (QOX) Step 1: quantize the solution space defined by two parents

into a finite number of points

)0.3,0.1(e

)0.1,0.3(g

x1 x2

[1.0, 3.0]

(1.0, 2.0, 3.0)

[1.0, 3.0]

(1.0, 2.0, 3.0)

QD

the number of levels

the number of variables

x1

x2 e

g

3.0

2.0

1.0

0.0 1.0 2.0 3.0

Y. W. Leung and Y. Wang, “An orthogonal genetic algorithm with quantization for global numerical optimization,” IEEE Transactions on Evolutionary Computation, vol. 5, no. 1, pp. 41-53, 2001.

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)0.3,0.1(e

)0.1,0.3(g

x1 x2

x1

x2 e

g

3.0

2.0

1.0

0.0 1.0 2.0 3.0 D=2

K=4


• Quantization orthogonal crossover (QOX) Step 2: select a small, but representative sample of points a

s the potential offspring by orthogonal design If D≤K, the first D columns of LM(QK) can be used directly If D>K, the decision vector will be divided into K subvectors

)0.3,0.8,0.0,0.1,0.5,0.2(=e

2H

3H

)0.5,0.2,0.6,0.2,0.3,0.4(=g

x1 x2 x4x3 x6x5

1H

4H

2.03.04.0

2.05.08.0

3.04.05.0

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• We proposed a generic framework for using QOX in DE variants

Y. Wang, Z. Cai, and Q. Zhang, “Enhancing the search ability of differential evolution through orthogonal crossover,” Information Sciences, vol. 185, no. 1, pp. 153-177, 2012.

1,Gx

2,Gx

,NP Gx

,k Gx

,k Gv

LM(QK)

_1,k Gu

_ 2,k Gu

_ ,k M Gu

,k Gu

comparison

Remark: Our framework uses QOX to complement binomial crossover or exponential crossover for searching some promising regions in the search

space.

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• An instantiation of our framework DE/rand/1/bin + QOX = OXDE

• Our framework can be easily generalized to other DE variants by replacing DE/rand/1/bin with other DE variants.

x1

x2 e

g

3.0

2.0

1.0

0.0 1.0 2.0 3.0

an improvement

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• The experimental results A suite of 24 test instances is used for our experimental studi

es the first 10 test instances are widely used in the evolutionary co

mputation community the other 14 test instances are the first 14 test instances designe

d for the IEEE CEC2005

Parameter settings NP=D=30, F=0.9, CR=0.9, and FESmax= 10,000×D

The average and standard deviation of the function error value were recorded for measuring the performance of each algorithm

For each test function, 50 independent runs were conducted

The t-test at a 0.05 significance level has been used in comparison

*| ( ) ( ) |bestf x f x

the best solution found by the algorithm in a run

the global optimum of the test function

59


• The experimental results How to measure the successful run

A run is successful if The parameter is set to 10-2 for test functions F6-F14, an

d 10-6 for the rest of the test functions

How to measure the convergence speed The mean and standard derivation of FESs among 50 inde

pendent runs are used to measure the convergence speed of an algorithm

*| ( ) ( ) |bestf x f x

successful condition

In a successful run, FESs is the number of FES needed for reaching successful condition

In an unsuccessful run, FESs is set to FESmax

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• The experimental results OXDE VS DE/rand/1/bin

OXDE can achieve at least one successful run on 11 test functions

DE/rand/1/bin can achieve at least one successful run on 7 test functions

OXDE has a faster convergence speed on these 11 test instances.

“ －” , “ ＋” , and “≈” denote that the performance of the corresponding algorithm is worse than, better than, and similar to that of OXDE, respectively.

OXDE certainly outperforms DE/rand/1/bin in terms of both the solution quality and the convergence speed.

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Effect of population size (NP=50, 100, 200, and 300)

OXDE performs significantly better than DE/rand/1/bin with different population sizes.


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Effect of population size (NP=50, 100, 200, and 300)

Fgrw (NP=100) F5 (NP=200)

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Effect of the number of variables (D=10, 50, 100, and 200)


The advantage of OXDE over DE/rand/1/bin increases as the number of variables increases.

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Effect of the number of variables (D=10, 50, 100, and 200)

Fack (D=100) Fsch (D=200)

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• The experimental results Runtime complexity of OXDE

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• The experimental results Can our framework improve other DE variants?

Our framework can greatly improve the performance of DEahcSPX, DE/rand/1/exp, DE/rand/2/exp, and DE/rand/2/bin

Our framework can also improve the performance of jDE, SaDE, and JADE to a certain degree

our framework could be an effective way to improve the performance of other DE variants.

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• The experimental results Comparison with opposition-based DE (ODE)


OXDE performs better than ODE.

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• The experimental results Comparison with other state-of-the-art EAs


OXDE is a generally good global function optimizer.

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• The experimental results Orthogonal crossover VS uniformly random sampling and H

alton sampling

the same level of improvement cannot be achieved via additional sampling strategies

OXDE is able to exhibit better performance than HSDE in nearly all test instances;URSDE shows better performance than OXDE in four test instances OXDE outperforms URSDE in eight test instances

70

Outline of My Talk





Conclusion

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Conclusion

• We have demonstrated that the experiences and knowledge obtained from the researchers can be exploited to improve the performance of DE significantly for the first time.

• We have verified that the search ability of DE can be enhanced by effectively probing the hyper-rectangle defined by the mutant and target vectors for the first time.

The source codes of CoDE and OXDE can be downloaded from the following URL: http://deptauto.csu.edu.cn/staffmember/YongWang.htm

http://deptauto.csu.edu.cn/staffmember/YongWang.htm

Recent Advances in Differential Evolution Yong Wang Lecturer, Ph.D. School of Information Science...

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