Test Functions Index

This page contains the general index of the benchmark problems used to test different Global Optimization algorithms. It also shows some statistics on the “difficulty” of a multi-modal test problem, based on the average successful minimization across all the Global Optimizers tested in this benchmark exercise.

The test suite currently contains:

1. 18 one-dimensional test functions with multiple local/global minima;
2. 184 multivariate problems (where the number of independent variables ranges from 2 to 17), again with multiple local/global minima.

Some of the multivariate problems are well known in the mathematical literature; for the less well-known functions, I have taken their definitions mostly from the following references:

• A Literature Survey of Benchmark Functions For Global Optimization Problems (http://arxiv.org/pdf/1308.4008v1.pdf)
• Test functions for optimization needs (http://www.zsd.ict.pwr.wroc.pl/files/docs/functions.pdf)
• A Numerical Evaluation of Several Stochastic Algorithms on Selected Continuous Global Optimization Test Problems (http://folk.uib.no/ssu029/Pdf_file/Ali05.pdf)

Test Functions “Hardness”

The test suite contains a variety of Global Optimization problems, some of them are harder to solve than others, irrespectively of the algorithm chosen to minimize the test function. The following table has been obtained by running all the Global Optimizers available against all the N-D test functions for a collection of 100 random starting points, and then averaging the successful minimizations across all the optimizers.

It can be easily seen that the DeVilliersGlasser02, Damavandi and CrossLegTable were always hard problems for all the algorithms, while others like Bukin02 and CosineMixture are amongst the easiest ones.

Benchmark test functions “hardness”

Optimization Method	N	Overall Success (%)
DeVilliersGlasser02	5	0.00
Damavandi	2	0.25
CrossLegTable	2	0.83
XinSheYang03	2	1.08
SineEnvelope	2	2.17
Whitley	2	4.92
Zimmerman	2	4.92
Griewank	2	6.08
Trefethen	2	6.58
Bukin06	2	6.83
Cola	17	7.50
CrownedCross	2	8.42
RosenbrockModified	2	8.42
Salomon	2	10.33
PowerSum	4	10.50
Gulf	3	10.92
NewFunction02	2	13.33
Mishra04	2	13.50
Stochastic	2	14.17
Kowalik	4	14.83
ZeroSum	2	15.08
DeVilliersGlasser01	4	16.92
Price02	2	18.00
EggHolder	2	18.92
Ripple01	2	19.25
Mishra03	2	20.42
XinSheYang01	2	20.92
NewFunction01	2	21.17
DropWave	2	21.25
Watson	6	21.58
Branin02	2	23.50
Schaffer01	2	23.67
Schaffer03	2	25.92
Easom	2	26.08
XinSheYang04	2	26.33
Tripod	2	27.33
Weierstrass	2	27.75
Deceptive	2	28.25
Schaffer04	2	28.75
XinSheYang02	2	31.33
OddSquare	2	31.75
Powell	4	32.58
Exp2	2	32.67
Paviani	10	33.33
Pathological	2	33.75
Schwefel06	2	34.67
HelicalValley	3	34.75
Shekel05	4	35.42
Wavy	2	37.00
WayburnSeader02	2	37.33
Shubert03	2	37.92
TestTubeHolder	2	37.92
Corana	4	38.08
YaoLiu09	2	38.42
Dolan	5	39.00
Shubert01	2	39.42
Rastrigin	2	39.50
Schaffer02	2	39.58
Chichinadze	2	40.33
Penalty02	2	41.08
Levy05	2	41.17
UrsemWaves	2	41.33
Price03	2	41.50
FreudensteinRoth	2	41.58
Xor	9	41.83
Hansen	2	41.92
Leon	2	42.50
Schwefel21	2	42.92
Trid	6	43.83
Rosenbrock	2	44.17
Langermann	2	45.67
Ripple25	2	46.33
Shekel10	4	47.83
Ackley	2	48.25
Hartmann6	6	48.50
ElAttarVidyasagarDutta	2	48.67
Shekel07	4	48.83
LennardJones	6	49.08
Ursem03	2	49.50
VenterSobiezcczanskiSobieski	2	49.58
Trigonometric02	2	49.58
Price04	2	49.83
Cube	2	49.92
BartelsConn	2	50.00
MieleCantrell	4	50.83
Mishra05	2	50.92
PermFunction01	2	51.00
BoxBetts	3	52.17
Mishra08	2	53.25
Decanomial	2	53.92
Shubert04	2	54.92
Mishra09	3	55.58
WayburnSeader01	2	55.83
JennrichSampson	2	56.25
Mishra06	2	57.08
Cigar	2	57.50
Step	2	58.25
Judge	2	58.33
Bird	2	59.00
Schwefel20	2	59.33
Schwefel22	2	59.42
DeflectedCorrugatedSpring	2	59.50
Hartmann3	3	59.83
Schwefel36	2	60.33
Quintic	2	62.67
Schwefel26	2	62.67
Rana	2	63.25
Penalty01	2	64.50
PermFunction02	2	64.75
EggCrate	2	64.92
Alpine01	2	65.17
GoldsteinPrice	2	65.50
YaoLiu04	2	65.67
Pinter	2	66.25
Beale	2	66.25
Levy13	2	66.33
Ursem04	2	66.42
StretchedV	2	68.00
Gear	4	68.08
Alpine02	2	68.17
Mishra10	2	68.33
Bukin04	2	68.92
RotatedEllipse01	2	69.25
Branin01	2	69.50
NeedleEye	2	69.58
Holzman	3	69.75
Mishra11	2	70.08
HimmelBlau	2	70.50
StyblinskiTang	2	70.50
Levy03	2	71.83
Colville	4	72.00
Matyas	2	72.00
NewFunction03	2	72.42
Qing	2	73.17
CrossInTray	2	74.08
Price01	2	74.33
RotatedEllipse02	2	74.92
Wolfe	3	75.00
CarromTable	2	75.17
Mishra07	2	75.33
Deb02	2	75.42
Deb01	2	75.50
Zirilli	2	76.58
Treccani	2	78.08
PenHolder	2	78.42
ThreeHumpCamel	2	78.83
Schwefel04	2	79.75
HolderTable	2	80.08
Parsopoulos	2	80.50
DeckkersAarts	2	81.17
Michalewicz	2	81.25
Sargan	2	81.25
McCormick	2	81.50
DixonPrice	2	81.58
MultiModal	2	81.67
Bohachevsky	2	81.75
Brent	2	81.92
Zettl	2	81.92
Adjiman	2	82.08
Schwefel01	2	82.08
Brown	2	82.17
SixHumpCamel	2	82.17
Zacharov	2	82.33
Sphere	2	82.75
Schwefel02	2	83.00
Quadratic	2	83.17
Mishra01	2	83.83
Csendes	2	84.17
Infinity	2	84.67
Ursem01	2	84.67
Giunta	2	85.00
Vincent	2	85.00
Hosaki	2	85.00
Exponential	2	85.33
SchmidtVetters	3	89.25
Plateau	2	89.42
Trigonometric01	2	90.83
Katsuura	2	91.00
Sodp	2	91.25
Mishra02	2	91.67
Keane	2	92.83
AMGM	2	95.17
Bukin02	2	100.00
CosineMixture	2	100.00

Multidimensional Test Functions Index

• N-D Test Functions A
  1. Ackley
  2. Adjiman
  3. Alpine01
  4. Alpine02
  5. AMGM

---

• N-D Test Functions B
  1. BartelsConn
  2. Beale
  3. Bird
  4. Bohachevsky
  5. BoxBetts
  6. Branin01
  7. Branin02
  8. Brent
  9. Brown
  10. Bukin02
  11. Bukin04
  12. Bukin06

---

• N-D Test Functions C
  1. CarromTable
  2. Chichinadze
  3. Cigar
  4. Cola
  5. Colville
  6. Corana
  7. CosineMixture
  8. CrossInTray
  9. CrossLegTable
  10. CrownedCross
  11. Csendes
  12. Cube

---

• N-D Test Functions D
  1. Damavandi
  2. Deb01
  3. Deb02
  4. Decanomial
  5. Deceptive
  6. DeckkersAarts
  7. DeflectedCorrugatedSpring
  8. DeVilliersGlasser01
  9. DeVilliersGlasser02
  10. DixonPrice
  11. Dolan
  12. DropWave

---

• N-D Test Functions E
  1. Easom
  2. EggCrate
  3. EggHolder
  4. ElAttarVidyasagarDutta
  5. Exp2
  6. Exponential

---

• N-D Test Functions F
  1. FreudensteinRoth

---

• N-D Test Functions G
  1. Gear
  2. Giunta
  3. GoldsteinPrice
  4. Griewank
  5. Gulf

---

• N-D Test Functions H
  1. Hansen
  2. Hartmann3
  3. Hartmann6
  4. HelicalValley
  5. HimmelBlau
  6. HolderTable
  7. Holzman
  8. Hosaki

---

• N-D Test Functions I
  1. Infinity

---

• N-D Test Functions J
  1. JennrichSampson
  2. Judge

---

• N-D Test Functions K
  1. Katsuura
  2. Keane
  3. Kowalik

---

• N-D Test Functions L
  1. Langermann
  2. LennardJones
  3. Leon
  4. Levy03
  5. Levy05
  6. Levy13

---

• N-D Test Functions M
  1. Matyas
  2. McCormick
  3. Michalewicz
  4. MieleCantrell
  5. Mishra01
  6. Mishra02
  7. Mishra03
  8. Mishra04
  9. Mishra05
  10. Mishra06
  11. Mishra07
  12. Mishra08
  13. Mishra09
  14. Mishra10
  15. Mishra11
  16. MultiModal

---

• N-D Test Functions N
  1. NeedleEye
  2. NewFunction01
  3. NewFunction02
  4. NewFunction03

---

• N-D Test Functions O
  1. OddSquare

---

• N-D Test Functions P
  1. Parsopoulos
  2. Pathological
  3. Paviani
  4. Penalty01
  5. Penalty02
  6. PenHolder
  7. PermFunction01
  8. PermFunction02
  9. Pinter
  10. Plateau
  11. Powell
  12. PowerSum
  13. Price01
  14. Price02
  15. Price03
  16. Price04

---

• N-D Test Functions Q
  1. Qing
  2. Quadratic
  3. Quintic

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• N-D Test Functions R
  1. Rana
  2. Rastrigin
  3. Ripple01
  4. Ripple25
  5. Rosenbrock
  6. RosenbrockModified
  7. RotatedEllipse01
  8. RotatedEllipse02

---

• N-D Test Functions S
  1. Salomon
  2. Sargan
  3. Schaffer01
  4. Schaffer02
  5. Schaffer03
  6. Schaffer04
  7. SchmidtVetters
  8. Schwefel01
  9. Schwefel02
  10. Schwefel04
  11. Schwefel06
  12. Schwefel20
  13. Schwefel21
  14. Schwefel22
  15. Schwefel26
  16. Schwefel36
  17. Shekel05
  18. Shekel07
  19. Shekel10
  20. Shubert01
  21. Shubert03
  22. Shubert04
  23. SineEnvelope
  24. SixHumpCamel
  25. Sodp
  26. Sphere
  27. Step
  28. Stochastic
  29. StretchedV
  30. StyblinskiTang

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• N-D Test Functions T
  1. TestTubeHolder
  2. ThreeHumpCamel
  3. Treccani
  4. Trefethen
  5. Trid
  6. Trigonometric01
  7. Trigonometric02
  8. Tripod

---

• N-D Test Functions U
  1. Ursem01
  2. Ursem03
  3. Ursem04
  4. UrsemWaves

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• N-D Test Functions V
  1. VenterSobiezcczanskiSobieski
  2. Vincent

---

• N-D Test Functions W
  1. Watson
  2. Wavy
  3. WayburnSeader01
  4. WayburnSeader02
  5. Weierstrass
  6. Whitley
  7. Wolfe

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• N-D Test Functions X
  1. XinSheYang01
  2. XinSheYang02
  3. XinSheYang03
  4. XinSheYang04
  5. Xor

---

• N-D Test Functions Y
  1. YaoLiu04
  2. YaoLiu09

---

• N-D Test Functions Z
  1. Zacharov
  2. ZeroSum
  3. Zettl
  4. Zimmerman
  5. Zirilli

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Univariate Test Functions Index

• 1-D Test Functions
  1. Problem02
  2. Problem03
  3. Problem04
  4. Problem05
  5. Problem06
  6. Problem07
  7. Problem08
  8. Problem09
  9. Problem10
  10. Problem11
  11. Problem12
  12. Problem13
  13. Problem14
  14. Problem15
  15. Problem18
  16. Problem20
  17. Problem21
  18. Problem22