APPENDIX I

Test Functions

205

In order to determine how well an optimization algorithm works, a variety oftest functions have been used as a check. We’ve listed 16 in this appendix. Ineach case we give a general form of the function, plot its value in one or twodimensions, give the global optimum in one or two dimensions, and list thedomain. Some of the functions are generalizable to N dimensions. MATLABcode for these functions appear in Appendix II. Some of the research intodeveloping test functions is reported in the references.

BIBLIOGRAPHY

De Jong, K. A. 1975. Analysis of the behavior of a class of genetic adaptive systems.Ph.D. Dissertation. University of Michigan, Ann Arbor.

Michalewicz, Z. 1992. Genetic Algorithms + Data Structures = Evolution Programs.New York: Springer-Verlag.

Schwefel, H. 1995. Evolution and Optimum Seeking. New York: Wiley.Whitley, D., K. Mathias, S. Rana, and J. Dzubera. 1996. Evaluating Evolutionary

Algorithms. Artificial Intelligence Journal 85:1–32.Whitley, D., R. Beveridge, C. Graves, and K. Mathias. 1995. Test Driving Three 1995

Genetic Algorithms: New Test Functions and Geometric Matching. Journal ofHeuristics 1:77–104.

Practical Genetic Algorithms, Second Edition, by Randy L. Haupt and Sue Ellen Haupt.ISBN 0-471-45565-2 Copyright © 2004 John Wiley & Sons, Inc.

206 TEST FUNCTIONS

x x

f

x

+ ( )

( ) =

- • £ £ •

cos

minimum:

for

0 1

F1

x x

f

x

+ ( )

( ) =

- • £ £ •

sin

minimum: 0

for

0

F2

x

f

x

nn

N2

1

1

=Â

( ) =

- • £ £ •

minimum: 0,0

for

F3

100 1

0

12 2 2

1

1

x x x

f

x

n n nn

N

n

+=

-

-[ ] + -[ ]{ }( ) =

- • £ £ •

Â

minimum: 1,1

for

F4

TEST FUNCTIONS 207

x x

f x x

x

nn

N

n

n

=Â - ( )

( ) = =

- • £ £ •

1

10 10

1 0

cos

minimum: at

for

F5

x x y y

f

x y

sin . sin

. , . .

,

4 1 1 2

0 9039 0 8668 18 5547

0 10

( ) + ( )( ) = -

£ £

minimum:

for

F7

x x x

f

x

2

9 6204 100 22

10 10

+( ) ( )( ) = -

- £ £

cos

. .minimum:

for

F6

y x x y

f

x y

sin . sin

. , . .

,

4 1 1 2

0 9039 0 8668 18 5547

0 10

( ) + ( )( ) = -

£ £

minimum:

for

F8

208 TEST FUNCTIONS

nx N

x

nn

N

n4

1

0 1=

ÂÈÎÍ

˘˚̇

+ ( )

- • £ £ •

,

minimum: varies

for

F9

14000

0

2

11

+ - ( )

( ) =

- • £ £ •

==’Â x

x

f

x

nn

n

N

n

N

n

cos

minimum: 0,0

for

F11

10 10 2

0

2

1

N x x

f

x

n nn

N

n

+ - ( )[ ]

( ) =

- • £ £ •

=Â cos p

minimum: 0,0

for

F10

0 50 5

1 0 1

0 5231

2 2 2

2 2.

sin ..

.

,

++ -

+ +( )( ) = -

- • £ £ •

x yx y

f

x y

minimum: 1.897,1.006

for

F12

TEST FUNCTIONS 209

x y x y x y

f

x y

2 2 0 25 2 20 1

30 0 5

0 0 0

+( ) +( ) +[ ]{ } + +

( ) =

- • £ £ •

. .sin .

,

,

minimum:

for

F13

J x y x y

f

x y

02 2 0 11 0 11

1 1 6606 0 3356

+( ) + - + -

( ) = -

- • £ £ •

. .

, . .

,

minimum:

for

F14

-

- -( ) = -

- £ £

- + + +( )e

f

x y

x y x y0 2 3 2 22 2

2 7730 5 16 947

5 5

. cos sin

. , .

,

minimum:

for

F15

- - +( )( ) - +( ) + +( )- -( ) = -

- £ £

x x y y y x

f

x y

sin 9 9 sin 0.5 9

minimum: 14.58, 20 23.806

for 20 , 20

F16