GA With a New Maximal Information Coefficient Based Mutation-libre

8/12/2019 GA With a New Maximal Information Coefficient Based Mutation-libre

1/6

New Genetic Algorithm with a Maximal InformationCoefficient Based Mutation

Nicholas Romito

ABSTRACT

In this paper, the Maximal Information Coefficient (MIC)will be used to modify the Genetic Algorithm (GA) in or-der to solve multi-variable optimization problems more ef-ficiently and accurately. The MIC modified GA (MICGA)learns the problem structure by calculating the MIC. Theoriginal GA is compared to the MICGA and many othertypes of optimization algorithms to determine the most ef-ficient optimization method.

Categories and Subject Descriptors

H.4 [Information Systems Applications]: Miscellaneous;D.2.8 [Software Engineering]: Metricscomplexity mea-sures, performance measures

General Terms

Theory

Keywords

Genetic Algorithm, MIC

1. INTRODUCTIONThere are many optimization problems that occur in the

world that need solving in all fields of study. Some of theseproblems, for example, include aligning MR and CT modal-ity images of a brain to create one composite image [1],optimizing the thermal management and package design ofa collector-up heterojunction bipolar transistor to improvemobile phones [12], or optimizing the system reliability andcost of a product [10]. All of these optimization problemsare generally very difficult to solve by hand; therefore, thebest way to solve these optimization problems is to use anefficient and accurate approximation tool.

Optimization problems usually have an objective functionfor which the global optimum (maximum or minimum) valuemust be determined. The minimum value of an objective

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission and/or a fee.

ACM Southeast13, April 4-6, 2013, Savannah, GACopyright 2013 ACM 978-1-4503-1901-0/13/04 ...$15.00.

function is the smallest valuey for y = f(x) where f(x) isthe objective function and x is the vector of independentvariables of size n depending on the dimension of the opti-mization problem.

The Genetic Algorithm (GA) [2] is an approximation al-gorithm that helps deal with optimization problems such asthe problems mentioned earlier, but as these problems growmore complex, faster, more accurate algorithms are needed

to solve these optimization problems. This paper proposesthat an improved real-coded GA can be implemented bylearning the problem structure to better solve the problem.

There are many other optimization algorithms that havebeen developed, and a brief description for each of the algo-rithms that will be tested against the Maximal InformationCoefficient modified GA (MICGA) proposed in this papercan be found in [8]. The MICGA is another variation ofthe simple GA and improves it by adapting the algorithmto each specific problem being solved.

2. COMPONENTS OF THE MICGA

2.1 The Simple GAThe simple GA uses the principles of evolution and natural

selection to determine the global minimum or maximum of agiven objective function. The genetic operators in the simpleGA are mutation and crossover, which cause the solutions toevolve and become better fitted to the objective functionsmaximum or minimum value.

Next, begins the selection of mates from the population,and in this paper, we will assume minimization is being per-formed. To start the evolution process, two individuals fromthe population are selected at random and are used as matesfor reproduction in order to produce better fitting solutionsto the problem.

After selection, reproduction occurs, which consists of twosteps: crossover and mutation. Crossover is the first geneticoperator to be performed on the mates that were selected.

Crossover takes some traits from both parents and com-bines them to create two new children. This genetic oper-ator is used to create solutions with better fitness to theobjective function. Whether or not crossover occurs de-pends on the user-defined crossover probability. If there isa crossover, each variable then has a fifty percent chance ofhaving crossover performed on it. Once a variable, xi, hasbeen selected to have crossover performed on it, the averagevalue of the two parent individuals xi is determined, andthe average value is then used for the childrens xi value.The xi value for each child is then offset by a small random


2/6

number, so they will not end with the same value in order toimprove search results. The final value forxi for each childis determined by:

child1 = Mean+.5Diff erencechild2 = Mean.5Diff erence

(1)

=

.999999 , > .999999

( .51

)1

3 , .5 < .999999

(2)1

3 , 0 < .50 , otherwise

(2)

= (1 .5

(1 + 2 DistanceDifference

)3)u (3)

where Difference is the difference between the two par-ent values ofxi, Distance is the distance from the parentsvalue ofxi to xis closest boundary, andu is a uniform, ran-dom variable. Once each variable has been visited and hadcrossover performed on it or not, the children are added tothe new population and mutation begins.

Just like in human evolution, there are some differencesbetween parents traits and childrens traits. This is caused

by the second genetic operator, mutation. This genetic op-erator is used to fine tune solutions or to force a solutionout of a local minimum value. Mutation runs through eachvariable of all the individuals and mutates them by offset-ting them with a small random number. Once a variable,xi, has been chosen for mutation, the variable will be offsetusing the following equations:

xinew =

low , xi+ < lowxi+ , low xi+ up

up , xi+ > up(4)

= (uplow)

u , u 11109

1

101 , .5< u


3/6

Table 1: Number of Function Evaluations for the Benchmark Functions in Table 2Function Simple GA MICGA RCCRO GA FEP CEP FES CES PSO GSO RCBBO DE CMAES G3PCX

f 1 15 000 0 150 00 0 1 50 000 15 000 0 150 00 0 1 50 000 1 500 00 1 500 00 150 00 0 15 00 00 15 00 00 1 50 000 150 000 1 50 000f 2 15 000 0 150 00 0 1 50 000 15 000 0 200 00 0 2 00 000 2 000 00 2 000 00 150 00 0 15 00 00 20 00 00 1 50 000 150 000 1 50 000f 3 15 000 0 150 00 0 2 50 000 25 000 0 500 00 0 5 00 000 5 000 00 5 000 00 250 00 0 25 00 00 50 00 00 2 50 000 250 000 2 50 000f 4 15 000 0 150 00 0 1 50 000 15 000 0 500 00 0 5 00 000 5 000 00 5 000 00 150 00 0 15 00 00 50 00 00 1 50 000 150 000 1 50 000f5 150000 150000 150000 150000 2000000 2000000 2000000 2000000 150000 150000 500000 150000 150000 150000f 6 15 000 0 150 00 0 1 50 000 15 000 0 150 00 0 1 50 000 1 500 00 1 500 00 150 00 0 15 00 00 15 00 00 1 50 000 150 000 1 50 000f 7 15 000 0 150 00 0 1 50 000 15 000 0 300 00 0 3 00 000 3 000 00 3 000 00 150 00 0 15 00 00 30 00 00 1 50 000 150 000 1 50 000f 8 15 000 0 150 00 0 1 50 000 15 000 0 900 00 0 9 00 000 9 000 00 9 000 00 150 00 0 15 00 00 30 00 00 1 50 000 150 000 1 50 000f 9 15 000 0 150 00 0 2 50 000 25 000 0 500 00 0 5 00 000 5 000 00 5 000 00 250 00 0 25 00 00 30 00 00 2 50 000 250 000 2 50 000

f1 0 15 000 0 150 00 0 1 50 000 15 000 0 150 00 0 1 50 000 1 500 00 1 500 00 150 00 0 1 500 00 15 00 00 1 50 000 150 000 1 50 000f1 1 15 000 0 150 00 0 1 50 000 15 000 0 200 00 0 2 00 000 2 000 00 2 000 00 150 00 0 1 500 00 30 00 00 1 50 000 150 000 1 50 000f1 2 15 000 0 150 00 0 1 50 000 15 000 0 150 00 0 1 50 000 1 500 00 1 500 00 150 00 0 1 500 00 15 00 00 1 50 000 150 000 1 50 000f1 3 15 000 0 150 00 0 1 50 000 15 000 0 150 00 0 1 50 000 1 500 00 1 500 00 150 00 0 1 500 00 15 00 00 1 50 000 150 000 1 50 000f14 150000 150000 7500 7500 10000 10000 10000 10000 7500 7500 10000 7500 7500 7500f1 5 15 000 0 150 00 0 2 50 000 25 000 0 400 00 0 4 00 000 4 000 00 4 000 00 250 00 0 2 500 00 10 00 00 2 50 000 250 000 2 50 000f16 150000 150000 1250 1250 10000 10000 10000 10000 1250 1250 10000 1250 1250 1250f17 150000 150000 5000 5000 10000 10000 10000 10000 5000 5000 10000 5000 5000 5000f18 150000 150000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000f19 150000 150000 4000 4000 10000 10000 10000 10000 4000 4000 10000 4000 4000 4000f20 150000 150000 7500 7500 20000 20000 20000 20000 7500 7500 20000 7500 7500 7500f21 150000 150000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000f22 150000 150000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000f23 150000 150000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000

Figure 2: The average MIC values for symmetric

and asymmetric functions calculated from the sim-

ple GA

4. SIMULATION RESULTS

4.1 Benchmark FunctionsThe MICGA and simple GA were tested with a set of

23 benchmark problems which can be seen in Table 2. Thistable includes the category of the function, the function itselffi, the name of the function, the number of variables usedin the function, the domain in which the solution liesS, andthe minimum value of the function fmin. These 23 functionsare standard benchmark functions for testing optimizationalgorithms, and Table 2 has been adopted from [8]. The

different categories of benchmark functions are as follows:

4.1.1 Unimodal Functions

f1 f7 are unimodal which means they do not have anylocal minima, so these functions are rather simple in terms offinding the global minimum value. This particular group isalso high-dimensional to increase the difficulty of the searchfor the global minimum.

4.1.2 High-Dimensional Multimodal Functions

f8 f13 are the high-dimensional multimodal functionswhich have many variables and have multiple local minima,

so the search for the global minimum, in this category, isvery difficult in relation to the other benchmark functions.

4.1.3 Low-Dimensional Multimodal Functions

f14 f23 are low-dimensional and multimodal, so theystill have multiple local minima similar to the second groupof functions, but there are less local minima than in theprevious multimodal group.

4.2 Experimental SettingAll of the simulations were ran on a personal computer

with an Intel Quad Core 3.20 GHz CPU and 5.6 GiB ofRAM. The MICGA was developed by modifying the simpleGA C language source code that can be found at [2].

The parameters used for the simple GA and MICGA wereprobability of mutationP(mutate) = .2, probability of crossover

P(xover) = .9, population sizeN P = 100, number of gener-ations maxgen = 1500, simulated binary crossover param-eter (SBX) SBX= 2, and polynomial mutation parameterP M= 100. The simple GA and MICGA both used 150000function evaluations for each of the benchmark functions inTable 2, and their average solution out of 100 runs was usedto compare to the other algorithms in Tables 3 - 5. Eachalgorithm was ranked based on its average solution, and anyties were broken using the standard deviation for those al-gorithms involved. Once the rank was determined for eachalgorithm in a category, the average rank of the algorithmswas calculated, and the algorithms final ranking came fromthe ordering of these average ranks.

4.3 ComparisonThe simple GA and MICGA were tested along with all

of the algorithms mentioned in Section 1. The data for theother algorithms were adopted from [8], and the simple GAand MICGA were tested as previously described in this sec-tion. The results are separated into the three categories offunctions, which were described earlier in this section.

4.3.1 Unimodal Functions

Table 3 shows that the MICGA performed poorly on thesefunctions relative to the other algorithms, but it still rankedtwo positions higher than the simple GA, which ranked lastout of all the algorithms.


4/6

Table 2: 23 Benchmark FunctionsCategory Test Function Name n S fmin

I f1 =ni=1

x2i Sphere Mo del 30 [100, 100]n 0

f2 =ni=1

|xi|+ni=1

|xi| Schwefelsproblem 2.22

30 [10, 10]n 0

f3 =ni=1

(i

j=1

xj)2 Schwefels

problem 1.230 [100, 100]n 0

f4 = maxi{|xi|, 1 i n} Schwefels

problem 2.21

30 [100, 100]n 0

f5 =n1i=1

(100(xi+1x2i )

2 + (xi1)2) Generalized

Rosenbrocksfunction

30 [30, 30]n 0

f6 =ni=1

(xi+ 0.5)2 Step function 30 [100, 100]n 0

f7 =ni=1

ix4i +random[0, 1) Quartic func-tion withnoise

30 [1.28, 1.28]n 0

II f8 = ni=1

(xisin(|xi|)) Generalized

Schwefelsproblem 2.26

30 [500, 500]n -12569.5

f9 =ni=1

(x2i 10cos(2xi) + 10) GeneralizedRastriginsfunction

30 [5.12, 5.12]n 0

f10 = 20exp0.2

1n

ni=1

x2i

exp

1n

ni=1

cos(2xi)

+ 20 +e

Ackleys func-tion

30 [32, 32]n 0

f11= 14000

ni=1

x2i ni=1

cos( xii

) + 1 GeneralizedGriewankfunction

30 [600, 600]n 0

f12 =

n{10 sin2(y1) +

29i=1

(yi

1)2[1 + 10sin2(yi+1) ] + (yn 1)2} +

30i=1

u(xi, 10, 100, 4)

Generalizedpenalizedfunctions 1

30 [50, 50]n 0

yi= 1 + 14

(xi+ 1)

u(xi, a , k , m) =

k(xia)m, xi > a

0, a xi a

k(xia)m, xi < a

f13 = 0.1{sin2(3x1) +

29i=1

(xi 1)2[1 +

sin2(3xi+1)]+(xn1)2[1+sin2(2x30)]}+

30i=1

u(xi, 5, 100, 4)

Generalizedpenalizedfunctions 2

30 [50, 50]n 0

III f14=

1500

+25j=1

1

j+2

i=1(xiaij)6

1

Shekels Fox-holes function

2 [65.536, 65.536]n 1

f15=11i=1

ai

x1(b2+bix2)

b2i+bix3+x4

2Kowaliksfunction

4 [5, 5]n 0.0003075

f16= 4x212.1x

41 +

13

x61 + x1x24x22 + 4x

42 Six-hump

camel-backfunction

2 [5, 5]n -1.0316285

f17 = x2 5.142 x21+ 5x162 +10

1 18

cos(x1) + 10

Branin func-

tion

2 [5, 10][0, 15] .398

f18= [1+ (x1 + x2 + 1)2(1914x1 + 3x

21

14x2+6x1x2+3x22)][30+(2x13x2)

2(1832x1+ 12x

21+ 48x236x1x2+ 27x

22)]

Goldstein-Price function

2 [2, 2]n 3

f19= 4

i=1

ciexp

4j=1

aij(xj pij)2

Hartmans

family func-tion 1

3 [0, 1]n -3.86

f20= 4

i=1

ciexp

6j=1

aij(xj pij)2

Hartmans

family func-tion 2

6 [0, 1]n -3.32

f21= 5

i=1

[(xai)(xai)T +ci]

1 Shekels fam-ily function 1

4 [0, 10]n -10

f22= 7

i=1

[(xai)(xai)T +ci]

1 Shekels fam-ily function 2

4 [0, 10]n -10

f22=

10i=1[(xai)(xai)

T

+ci]1

Shekels fam-ily function 3 4 [0, 10]n

-10


5/6

Table 3: Simulation Results for f1f7Simple GA MICGA RCCRO1 GA FEP CEP FES CES PSO GSO RCBBO DE CMAES G3PCX

f1 Mean 8.442E+01 6.625E-01 6.427E-07 3.171E+00 5.700E-04 2.200E-04 2.500E-04 3.400E-05 3.693E-37 1.948E-08 1.390E-03 6.576E-06 6.093E-29 6.404E-79Std Dev 1.362E+01 4.976E-01 2.099E-07 1.662E+00 1.300E-04 5.900E-04 6.800E-04 8.600E-06 2.460E-36 1.163E-08 5.500E-04 1.132E-06 1.554E-29 1.248E-78

Rank 14 12 5 13 10 8 9 7 2 4 11 6 3 1f2 Mean 3.510E+00 1.616E-01 2.196E-03 5.771E-01 8.100E-03 2.600E-03 6.000E-02 2.100E-02 2.917E-24 3.704E-05 7.990E-02 2.894E-04 3.480E-14 2.803E+01

Std Dev 4.089E-01 8.200E-02 4.341E-04 1.306E-01 7.700E-04 1.700E-04 9.600E-03 2.200E-03 1.136E-23 8.619E-05 1.440E-02 2.518E-05 4.034E-15 1.012E+01Rank 13 11 5 12 7 6 9 8 1 3 10 4 2 14

f3 Mean 1.700E+308 1.700E+308 2.966E-07 9.750E+03 1.600E-02 5.000E-02 1.400E-03 1.300E-04 1.198E-03 5.783E+00 2.270E+01 1.212E+04 1.511E-26 1.064E-76Std Dev 0.000E+00 0.000E+00 1.146E-07 2.595E+03 1.400E-02 6.600E-02 5.300E-04 8.500E-05 2.111E-03 3.681E+00 1.030E+01 1.554E+03 3.644E-27 1.532E-76

Rank 13 13 3 11 7 8 6 4 5 9 10 12 2 1f4 Mean 5.224E+00 8.237E-01 9.318E-03 7.961E+00 3.000E-01 2.000E+00 5.500E-03 3.500E-01 4.123E-01 1.075E-01 3.090E-02 5.790E+00 3.994E-15 4.543E+01

Std Dev 6.053E-01 5.110E-01 3.657E-03 1.506E+00 5.000E-01 1.200E+00 6.500E-04 4.200E-01 2.500E-01 3.998E-02 7.270E-03 4.559E-01 5.311E-16 8.092E+00Rank 11 9 3 13 6 10 2 7 8 5 4 12 1 14

f5 Mean 1.695E+03 7.243E+01 2.706E+01 3.386E+02 5.060E+00 6.170E+00 3.328E+01 6.690E+00 3.736E+01 4.984E+01 5.540E+01 9.338E+01 5.581E-01 3.091E+00Std Dev 3.318E+02 7.203E+01 3.427E+01 3.615E+02 5.870E+00 1.361E+01 4.313E+01 1.445E+01 3.214E+01 3.018E+01 3.520E+01 1.734E+01 1.390E+00 1.639E+01

Rank 14 11 6 13 3 4 7 5 8 9 10 12 1 2f6 Mean 8.656E+01 1.260E+00 0.000E+00 3.697E+00 0.000E+00 5.778E+02 0.000E+00 4.112E+02 1.460E-01 1.600E-02 0.000E+00 0.000E+00 7.000E-02 9.462E+01

Std Dev 1.234E+01 1.760E+00 0.000E+00 1.952E+00 0.000E+00 1.126E+03 0.000E+00 6.954E+02 4.182E-01 1.333E-01 0.000E+00 0.000E+00 2.932E-01 5.969E+01Rank 11 9 1 10 1 14 1 13 8 6 1 1 7 12

f7 Mean 8.802E+00 8.671E+00 5.405E-03 1.045E-01 7.600E-03 1.800E-02 1.200E-02 3.000E-02 9.902E-03 7.377E-02 1.750E-02 3.967E-02 2.209E-01 9.797E-01Std Dev 9.635E-01 9.437E-01 2.985E-03 3.622E-02 2.600E-03 6.400E-03 5.800E-03 1.500E-02 3.538E-02 9.256E-02 6.430E-03 7.832E-03 8.653E-02 4.627E-01

Rank 14 13 1 10 2 6 4 7 3 9 5 8 11 12Average Rank 12.857 11.143 3.429 11.714 5.143 8.000 5.429 7.286 5.000 6.429 7.286 7.857 3.857 8.000

Overall Rank 14 12 1 13 4 10 5 7 3 6 8 9 2 11

Table 4: Simulation Results for f8 f13Simple GA MICGA RCCRO1 GA FEP CEP FES CES PSO GSO RCBBO DE CMAES G3PCX

f8 Mean -1.233E+04 -1.257E+04 -1.26E+04 -1.257E+04 -1.255E+04 -7.917E+03 -7.550E+03 -7.550E+03 -9.660E+03 -1.257E+04 -1.257E+04 -1.257E+04 -9.873E+07 -2.577E+03Std Dev 1.234E+03 1.257E+03 2.32E-02 2.109E+00 5.26E+01 6.345E+02 6.314E+02 6.314E+02 4.638E+02 2.214E-02 2.200E-05 2.330E-05 8.547E+08 4.126E+02

Rank 9 7 2 3 8 11 12 12 10 6 4 5 1 14f9 Mean 4.791E+01 1.369E+01 9.08E-04 6.509E-01 4.600E-02 8.900E+01 7.082E+01 7.082E+01 2.079E+01 1.018E+00 2.620E-02 7.261E-05 4.950E+01 1.740E+02

Std Dev 6.714E+00 3.784E+00 2.88E-04 3.594E-01 1.20E-02 2.310E+01 2.149E+01 2.149E+01 5.940E+00 9.509E-01 9.760E-03 3.376E-05 1.229E+01 3.199E+01Rank 9 7 2 5 4 13 11 11 8 6 3 1 10 14

f10 Mean 1.718E+00 1.718E+00 1.94E-03 8.678E-01 1.800E-02 9.200E+00 9.070E+00 9.070E+00 1.340E-03 2.655E-05 2.510E-02 7.136E-04 4.607E+00 1.352E+01Std Dev 1.718E-01 1.718E-01 4.19E-04 2.805E-01 2.10E-02 2.800E+00 2.840E+00 2.840E+00 4.239E-02 3.082E-05 5.510E-03 6.194E-05 8.725E+00 4.815E+00

Rank 8 8 4 7 5 13 11 11 3 1 6 2 10 14f 11 M ean 1.758E+00 7.250E- 01 1.117E- 02 1.004E+00 1.600E- 02 8.600E- 02 3.800E- 01 3.800E- 01 2.323E- 01 3.079E- 02 4.820E- 01 9.05E -05 7.395E- 04 1.127E- 02

S td D ev 2.012E- 01 1.774E- 01 1.622E- 02 6.755E- 02 2.20E -02 1.200E- 01 7.700E- 01 7.700E- 01 4.434E- 01 3.087E- 02 8.490E- 02 3.402E- 05 2.389E- 03 1.310E- 02Rank 14 12 3 13 5 7 9 10 8 6 11 1 2 4

f 12 M ean 7.116E- 01 1.44E -03 2.074E- 02 4.372E- 02 9.200E- 06 1.760E+00 1.180E+00 1.180E+00 3.950E- 02 2.765E- 11 3.280E- 05 1.886E- 04 5.167E- 03 4.593E+00Std Dev 1.564E-01 1.787E-03 5.485E-02 5.058E-02 6.14E-05 2.400E+00 1.870E+00 1.870E+00 9.142E-02 9.167E-11 3.330E-05 4.266E-08 7.338E-03 5.984E+00

Rank 10 5 7 9 2 12 11 11 8 1 3 4 6 14f13 Mean 2.236E+09 2.236E+09 7.048E-07 1.681E-01 1.600E-04 1.400E+00 1.390E+00 1.390E+00 5.05E-02 4.695E-05 3.720E-04 9.519E-07 1.639E-03 2.349E+01

Std Dev 6.038E+08 6.038E+08 5.901E-07 7.068E-02 7.30E-05 3.700E+00 3.330E+00 3.330E+00 5.691E-01 7.011E-04 4.630E-04 2.021E-07 4.196E-03 2.072E+01Rank 13 13 1 8 4 11 9 9 7 3 5 2 6 12

Average Rank 10.500 8.667 3.167 7.500 4.667 11.167 10.500 10.667 7.333 3.833 5.333 2.500 5.833 12.000

Overall Rank 10 9 2 8 4 13 11 12 7 3 5 1 6 14

Table 5: Simulation Results for f14f23Simple GA MICGA RCCRO1 GA FEP CEP FES CES PSO GSO RCBBO DE CMAES G3PCX

f14 Mean 1.652E+00 1.494E+00 9.980E-01 9.989E-01 1.220E+00 1.660E+00 1.200E+00 2.160E+00 1.024E+00 9.980E-01 9.980E-01 1.576E+00 1.246E+01 1.231E+01Std Dev 8.756E-01 4.730E-01 2.317E-02 4.433E-03 5.600E-01 1.190E+00 6.300E-01 1.820E+00 1.450E-01 0.000E+00 2.740E-05 2.140E+00 5.529E+00 5.882E+00

Rank 10 8 3 4 7 11 6 12 5 1 2 9 14 13f 15 M ean 4.211E- 02 4.213E- 02 9.077E- 04 7.088E- 03 5.000E- 04 4.700E- 04 9.700E- 04 1.200E- 03 3.807E- 04 3.771E- 04 7.860E- 04 5.372E- 04 6.554E- 04 5.332E- 04

S td D ev 4.211E- 03 4.213E- 03 2.876E- 04 7.855E- 03 3.200E- 04 3.000E- 04 4.200E- 04 1.600E- 05 2.509E- 04 2.597E- 04 1.800E- 04 1.221E- 04 3.730E- 04 3.784E- 04Rank 13 14 9 12 4 3 10 11 2 1 8 6 7 5

f16 Mean -1.032E+00 -1.032E+00 -1.032E+00 -1.030E+00 -1.030E+00 -1.030E+00 -1.032E+00 -1.032E+00 -1.014E+00 -1.032E+00 -1.031E+00 -1.019E+00 -1.015E+00 -4.928E-01Std Dev 1.032E-01 1.032E-01 4.843E-04 3.143E-03 4.900E-04 4.900E-04 6.000E-07 6.000E-07 1.279E-02 0.000E+00 9.010E-04 1.869E-02 4.148E-01 3.367E-01

Rank 5 5 4 10 8 8 2 2 13 1 7 11 12 14f 17 M ean 3.979E- 01 3.979E- 01 3.979E- 01 4.040E- 01 3.980E- 01 3.980E- 01 3.980E- 01 3.980E- 01 4.040E- 01 3.979E- 01 3.984E- 01 3.995E- 01 3.979E- 01 5.560E+01

Std Dev 3.979E-02 3.979E-02 8.525E-07 1.039E-02 1.500E-07 1.500E-07 6.000E-08 6.000E-08 6.881E+01 0.000E+00 6.770E-04 4.281E-03 1.047E-15 1.071E-13Rank 4 4 3 12 8 8 6 6 13 1 10 11 2 14

f18 Mean 3.053E+00 3.053E+00 3.001E+00 7.503E+00 3.020E+00 3.000E+00 3.000E+00 3.000E+00 3.005E+00 3.000E+00 3.010E+00 3.479E+00 5.700E+00 8.670E+00Std Dev 6.118E-01 6.118E-01 1.171E-03 1.040E+01 1.100E-01 0.000E+00 0.000E+00 0.000E+00 1.212E-03 0.000E+00 1.120E-02 3.319E+00 1.051E+01 1.290E+01

Rank 9 9 5 13 8 1 1 1 6 1 7 11 12 14f19 Mean -1.000E+00 -1.000E+00 -3.863E+00 -3.862E+00 -3.860E+00 -3.860E+00 -3.860E+00 -3.860E+00 -3.858E+00 -3.863E+00 -3.862E+00 -3.862E+00 -3.725E+00 -3.598E+00

S td D ev 1.000E- 01 1.000E- 01 1.464E- 03 6.284E- 04 1.400E- 02 1.400E- 02 4.000E- 03 1.400E- 05 3.213E- 03 3.843E- 06 3.650E- 04 1.672E- 03 5.744E- 01 1.869E- 01Rank 13 13 4 2 8 8 6 6 10 1 3 5 11 12

f20 Mean -1.000E+00 -1.000E+00 -3.319E+00 -3.263E+00 -3.270E+00 -3.280E+00 -3.230E+00 -3.240E+00 -3.185E+00 -3.270E+00 -3.317E+00 -3.316E+00 -3.290E+00 -1.980E-01S td D ev 1.000E- 01 1.000E- 01 2.115E- 03 6.040E- 02 5.900E- 02 5.800E- 02 1.200E- 01 5.700E- 02 6.105E- 02 5.965E- 02 2.360E- 02 6.674E- 03 5.305E- 02 4.327E- 01

Rank 12 12 1 8 6 5 10 9 11 7 2 3 4 14f21 Mean -7.096E+00 -7.097E+00 -1.011E+01 -5.165E+00 -5.520E+00 -6.860E+00 -5.540E+00 -6.960E+00 -7.544E+00 -6.090E+00 -5.513E+00 -8.739E+00 -6.683E+00 -7.476E-01

Std Dev 3.310E+00 3.310E+00 3.505E-02 2.925E+00 1.590E+00 2.670E+00 1.820E+00 3.100E+00 3.030E+00 3.456E+00 3.350E+00 1.571E+00 3.719E+00 3.170E-01Rank 5 4 1 13 11 7 10 6 3 9 12 2 8 14

f22 Mean -8.010E+00 -8.015E+00 -1.035E+01 -5.443E+00 -5.520E+00 -8.270E+00 -6.760E+00 -8.310E+00 -8.355E+00 -6.555E+00 -6.800E+00 -9.199E+00 -6.574E+03 -9.468E-01Std Dev 3.279E+00 3.278E+00 4.838E-02 3.278E+00 2.120E+00 2.950E+00 3.010E+00 3.100E+00 2.018E+00 3.244E+00 3.520E+00 1.217E+00 3.641E+00 3.761E-01

Rank 8 7 2 13 12 6 10 5 4 11 9 3 1 14f23 Mean -9.089E+00 -9.047E+00 -1.048E+01 -4.911E+00 -6.570E+00 -9.100E+00 -7.630E+00 -8.500E+00 -8.944E+00 -7.402E+00 -7.285E+00 -9.229E+00 -7.576E+00 -1.130E+00

Std Dev 2.915E+00 2.924E+00 3.885E-02 3.487E+00 3.140E+00 2.920E+00 3.270E+00 1.250E+00 1.630E+00 3.213E+00 3.380E+00 1.325E+00 3.741E+00 3.678E-01Rank 4 5 1 13 12 3 8 7 6 10 11 2 9 14

Average Rank 8.222 8.100 3.300 10.000 8.400 6.000 6.900 6.500 7.300 4.300 7.100 6.300 8.000 12.800

Overall Rank 11 10 1 13 12 3 6 5 8 2 7 4 9 14


6/6

Figure 3: The percentage of improvement of the

MICGA over the simple GA

4.3.2 High-Dimensional Multimodal Functions

Table 4 shows that the MICGA p erformed the b est in thiscategory over the other two categories. The MICGA still didnot rank very well compared to the other algorithms as itfinished with ranking of 9, but the simple GA still trailedthe MICGA by one rank.

4.3.3 Low-Dimensional Multimodal Functions

As can be seen in Table 5, the MICGA ranked 10 out ofthe 14 algorithms, so the MICGA still is not the best optionby any means. The simple GA had a final ranking of 11 outof 14 in this category, so the MICGA is still an improvementover the original algorithm.

4.4 Discussion

From the Tables 3 - 5, it can be seen that the MICGAis not the best optimization algorithm that can be used,but, from Figure 3, the MICGA is a significant improve-ment over the original simple GA. This improvement showsthat using the MIC to increase the accuracy of optimizationalgorithms is possible, and the application of the MIC toanother, more accurate algorithm could provide even betterresults than the algorithms in this paper. Also, the MICGAranked higher in the more difficult benchmark function cat-egory than in the simplest category, which leads to the con-clusion that the use of the MIC allows for a more consistentperformance across all problems.

It can be inferred that the most accurate algorithm, RC-CRO1, can be improved even further using the MIC, and

it can possibly be improved in the high-dimensional mul-timodal function category to the best algorithm since theMICGA performed the best in this category. It can be pre-dicted that the MIC will allow for consistency across eachcategory, so infusing the MIC with RCCRO1 could improvethe accuracy of RCCRO1 to be more accurate than any otheralgorithm in this paper.

5. CONCLUSIONAs real world optimization problems grow more difficult,

better algorithms are needed to solve these problems. Ifan algorithm can learn the structure of a problem, the algo-

rithm can find a significantly better solution to the problem.The MIC proved to be a viable option for determining thestructure of a function and notably improved the accuracy ofthe simple GA. Although this technique improved the accu-racy of the simple GA, it did not make this improved simpleGA (MICGA) a feasible option for optimization problemsbecause it did not perform as well as other algorithms.

Since the MIC successfully improved the accuracy of the

simple GA, it can be inferred that it will do the same forany other optimization algorithm; therefore, in the future,the MIC should be applied to the other algorithms in thispaper in order to determine if a more effective algorithm canbe created for solving optimization problems.

6. REFERENCES[1] A. Das and B. Mahua. Affine-based registration of ct

and mr modality images of human brain usingmultiresolution approaches: comparative study ongenetic algorithm and particle swarm optimization.Neural Computing & Applications, 20(2):223237,2011.

[2] K. Deb. Genetic algorithm [computer program], 2001.

[3] K. Deb, A. Anand, and D. Joshi. A computationalyefficient evolutionary algorithm for real-parameteroptimization. Evolutionary Computation, 10.

[4] W. Gong, Z. Cai, C. X. Ling, and H. Li. A real-codedbiogeography-based optimization with mutation, 2009.

[5] N. Hansen and A. Ostermeir. Completelyderandomized self-adaptation in evolution strategies.Evolutionary Computation, 9(2):159195, 2001.

[6] S. He, Q. Wu, and J. R. Saunders. Group searchoptimizer: An optimization algorithm inspired byanimal searching behavior. IEEE Transactions onEvolutionary Computation, 13(5):973990, 2009.

[7] J. Kennedy and R. Eberhart. Swarm Intelligence.Morgan Kaufmann, San Francisco, CA, 2001.

[8] A. Y. Lam, V. O. Li, and J. J. Yu. Real-codedchemical reaction optimization. IEEE Transactions onEvolutionary Computation, 16(3):339353, 2012.

[9] D. N. Reshef, Y. A. Reshef, H. K. Finucane, S. R.Grossman, G. McVean, P. J. Turnbaugh, E. S. Lander,M. Mitzenmacher, and P. C. Sabeti. Detecting novelassociations in large data sets.Science Magazine,334(6062):15181524, 2011.

[10] L. Sahoo, A. K. Bhunia, and P. K. Kapur. Geneticalgorithm based multi-objective reliabilityoptimization in interval environment. Computers &Industrial Engineering, 62(1):152160, 2012.

[11] R. Storn and K. Price. Differential evolution - a simpleand efficient heuristic for global optimization over

continuous spaces.Journal of Global Optimization,11(4):341359, 1997.

[12] H.-C. Tseng and J.-Y. Chen. Thermal analysis andpackaging optimization of collector-up hbts using anenhanced genetic-algorithm methodology. Packaging &Manufacturing Technology, 2(2):231239, 2012.

[13] X. Yao and Y. Liu. Faster evoultion strategies. InProc. 6th International Conference of EvolutionaryProgramming, pages 151162, 1997.

[14] X. Yao, Y. Liu, and G. Lin. Evolutionaryprogramming made faster.IEEE Transactions onEvolutionary Computation, 3(2):82102, 1999.

GA With a New Maximal Information Coefficient Based Mutation-libre

Documents

Transcript of GA With a New Maximal Information Coefficient Based Mutation-libre