# 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:

## 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
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
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
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
Schwefel01 2 82.08
Brown 2 82.17
SixHumpCamel 2 82.17
Zacharov 2 82.33
Sphere 2 82.75
Schwefel02 2 83.00
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