Test Problems

In the future, we are planning to provide a comprehensive overview of the problems. So far, we have managed to describe a view of them and plot the functions. Please note that we have used for some problems the descriptions from here.

We want to keep the function definitions as error-free as possible and matching with the implementation. If you find any deviation, please let us know.

Various test problems are already implemented and available by providing the corresponding problem name we have assigned to it. A couple of problems can be further parameterized by providing the number of variables, constraints, or other problem-dependent constants.

[1]:
from pymoo.factory import get_problem

p = get_problem("dtlz1_-1", n_var=20, n_obj=5)

# create a simple test problem from string
p = get_problem("Ackley")

# the input name is not case sensitive
p = get_problem("ackley")

# also input parameter can be provided directly
p = get_problem("dtlz1_-1", n_var=20, n_obj=5)

Many-Objective

Problem

Description

DTLZ1

many-objective

DTLZ2

DTLZ3

DTLZ4

DTLZ5

DTLZ6

DTLZ7

DTLZ1^-1

Scaled DTLZ

Convex DTLZ

WFG1

WFG2

WFG3

WFG4

WFG5

WFG6

WFG7

WFG8

WFG9

Multi-Objective

Problem

Variables

Objectives

Constraints

Description

BNH

2

2

2

Carside

7

3

10

Kursawe

3

2

OSY

6

2

6

TNK

2

2

2

Truss2D

3

2

1

Welded Beam

4

2

4

CTP1

2

2

s

CTP2

2

2

s

CTP3

2

2

s

CTP4

2

2

s

CTP5

2

2

s

CTP6

2

2

s

CTP7

2

2

s

CTP8

2

2

s

ZDT1

30

2

ZDT2

30

2

ZDT3

30

2

ZDT4

10

2

ZDT5

80

2

ZDT6

10

2

DASCMOP1

30

2

11

DASCMOP2

30

2

11

DASCMOP3

30

2

11

DASCMOP4

30

2

11

DASCMOP5

30

2

11

DASCMOP6

30

2

11

DASCMOP7

30

3

7

DASCMOP8

30

3

7

DASCMOP9

30

3

7

MW1

15

2

1

MW2

15

2

1

MW3

15

2

2

MW4

15

3

1

MW5

15

2

3

MW6

15

2

1

MW7

15

2

2

MW8

15

3

1

MW9

15

2

1

MW10

15

2

3

MW11

15

2

4

MW12

15

2

2

MW13

15

2

2

MW14

15

3

1

SymPart

2

2

OmniTest

s

2

MODAct

20

2-5

7-10

Real-world mechanical design problems

Single-Objective

Problem

Variables

Constraints

Type

Ackley

s

Cantilevered Beams

4

2

Griewank

s

Himmelblau

2

Knapsack

s

1

Pressure Vessel

4

4

Rastrigin

s

Rosenbrock

s

Schwefel

s

Sphere

s

Zakharov

s

G01

13

9

G02

20

2

G03

10

1

G04

5

6

G05

4

5

G06

2

2

G07

10

8

G08

2

2

G09

8

6

Global Optimization

Implementations are taken from (SciPy)[https://github.com/scipy/scipy/tree/master/benchmarks/benchmarks/go_benchmark_functions] where a variety of global optimization benchmark problems are provided. Pymoo uses a wrapper class to make all those functions available.

Problem

Variables

Constraints

Name

AMGM

2

“go-amgm”

Ackley01

2

“go-ackley01”

Ackley02

2

“go-ackley02”

Ackley03

2

“go-ackley03”

Adjiman

2

“go-adjiman”

Alpine01

2

“go-alpine01”

Alpine02

2

“go-alpine02”

BartelsConn

2

“go-bartelsconn”

Beale

2

“go-beale”

BiggsExp02

2

“go-biggsexp02”

BiggsExp03

3

“go-biggsexp03”

BiggsExp04

4

“go-biggsexp04”

BiggsExp05

5

“go-biggsexp05”

Bird

2

“go-bird”

Bohachevsky1

2

“go-bohachevsky1”

Bohachevsky2

2

“go-bohachevsky2”

Bohachevsky3

2

“go-bohachevsky3”

BoxBetts

3

“go-boxbetts”

Branin01

2

“go-branin01”

Branin02

2

“go-branin02”

Brent

2

“go-brent”

Brown

2

“go-brown”

Bukin02

2

“go-bukin02”

Bukin04

2

“go-bukin04”

Bukin06

2

“go-bukin06”

CarromTable

2

“go-carromtable”

Chichinadze

2

“go-chichinadze”

Cigar

2

“go-cigar”

Cola

17

“go-cola”

Colville

4

“go-colville”

Corana

4

“go-corana”

CosineMixture

2

“go-cosinemixture”

CrossInTray

2

“go-crossintray”

CrossLegTable

2

“go-crosslegtable”

CrownedCross

2

“go-crownedcross”

Csendes

2

“go-csendes”

Cube

2

“go-cube”

Damavandi

2

“go-damavandi”

DeVilliersGlasser01

4

“go-devilliersglasser01”

DeVilliersGlasser02

5

“go-devilliersglasser02”

Deb01

2

“go-deb01”

Deb03

2

“go-deb03”

Decanomial

2

“go-decanomial”

Deceptive

2

“go-deceptive”

DeckkersAarts

2

“go-deckkersaarts”

DeflectedCorrugatedSpring

2

“go-deflectedcorrugatedspring”

DixonPrice

2

“go-dixonprice”

Dolan

5

“go-dolan”

DropWave

2

“go-dropwave”

Easom

2

“go-easom”

Eckerle4

3

“go-eckerle4”

EggCrate

2

“go-eggcrate”

EggHolder

2

“go-eggholder”

ElAttarVidyasagarDutta

2

“go-elattarvidyasagardutta”

Exp2

2

“go-exp2”

Exponential

2

“go-exponential”

FreudensteinRoth

2

“go-freudensteinroth”

Gear

4

“go-gear”

Giunta

2

“go-giunta”

GoldsteinPrice

2

“go-goldsteinprice”

Griewank

2

“go-griewank”

Gulf

3

“go-gulf”

Hansen

2

“go-hansen”

Hartmann3

3

“go-hartmann3”

Hartmann6

6

“go-hartmann6”

HelicalValley

3

“go-helicalvalley”

HimmelBlau

2

“go-himmelblau”

HolderTable

2

“go-holdertable”

Hosaki

2

“go-hosaki”

Infinity

2

“go-infinity”

JennrichSampson

2

“go-jennrichsampson”

Judge

2

“go-judge”

Katsuura

2

“go-katsuura”

Keane

2

“go-keane”

Kowalik

4

“go-kowalik”

Langermann

2

“go-langermann”

LennardJones

6

“go-lennardjones”

Leon

2

“go-leon”

Levy03

2

“go-levy03”

Levy05

2

“go-levy05”

Levy13

2

“go-levy13”

Matyas

2

“go-matyas”

McCormick

2

“go-mccormick”

Meyer

3

“go-meyer”

Michalewicz

2

“go-michalewicz”

MieleCantrell

4

“go-mielecantrell”

Mishra01

2

“go-mishra01”

Mishra02

2

“go-mishra02”

Mishra03

2

“go-mishra03”

Mishra04

2

“go-mishra04”

Mishra05

2

“go-mishra05”

Mishra06

2

“go-mishra06”

Mishra07

2

“go-mishra07”

Mishra08

2

“go-mishra08”

Mishra09

3

“go-mishra09”

Mishra10

2

“go-mishra10”

Mishra11

2

“go-mishra11”

MultiModal

2

“go-multimodal”

NeedleEye

2

“go-needleeye”

NewFunction01

2

“go-newfunction01”

NewFunction02

2

“go-newfunction02”

OddSquare

2

“go-oddsquare”

Parsopoulos

2

“go-parsopoulos”

Pathological

2

“go-pathological”

Paviani

10

“go-paviani”

PenHolder

2

“go-penholder”

Penalty01

2

“go-penalty01”

Penalty02

2

“go-penalty02”

PermFunction01

2

“go-permfunction01”

PermFunction02

2

“go-permfunction02”

Pinter

2

“go-pinter”

Plateau

2

“go-plateau”

Powell

4

“go-powell”

PowerSum

4

“go-powersum”

Price01

2

“go-price01”

Price02

2

“go-price02”

Price03

2

“go-price03”

Price04

2

“go-price04”

Qing

2

“go-qing”

Quadratic

2

“go-quadratic”

Quintic

2

“go-quintic”

Rana

2

“go-rana”

Rastrigin

2

“go-rastrigin”

Ratkowsky01

4

“go-ratkowsky01”

Ratkowsky02

3

“go-ratkowsky02”

Ripple01

2

“go-ripple01”

Ripple25

2

“go-ripple25”

Rosenbrock

2

“go-rosenbrock”

RosenbrockModified

2

“go-rosenbrockmodified”

RotatedEllipse01

2

“go-rotatedellipse01”

RotatedEllipse02

2

“go-rotatedellipse02”

Salomon

2

“go-salomon”

Sargan

2

“go-sargan”

Schaffer01

2

“go-schaffer01”

Schaffer02

2

“go-schaffer02”

Schaffer03

2

“go-schaffer03”

Schaffer04

2

“go-schaffer04”

Schwefel01

2

“go-schwefel01”

Schwefel02

2

“go-schwefel02”

Schwefel04

2

“go-schwefel04”

Schwefel06

2

“go-schwefel06”

Schwefel20

2

“go-schwefel20”

Schwefel21

2

“go-schwefel21”

Schwefel22

2

“go-schwefel22”

Schwefel26

2

“go-schwefel26”

Schwefel36

2

“go-schwefel36”

Shekel05

4

“go-shekel05”

Shekel07

4

“go-shekel07”

Shekel10

4

“go-shekel10”

Shubert01

2

“go-shubert01”

Shubert03

2

“go-shubert03”

Shubert04

2

“go-shubert04”

SineEnvelope

2

“go-sineenvelope”

SixHumpCamel

2

“go-sixhumpcamel”

Sodp

2

“go-sodp”

Sphere

2

“go-sphere”

Step

2

“go-step”

Step2

2

“go-step2”

Stochastic

2

“go-stochastic”

StretchedV

2

“go-stretchedv”

StyblinskiTang

2

“go-styblinskitang”

TestTubeHolder

2

“go-testtubeholder”

ThreeHumpCamel

2

“go-threehumpcamel”

Thurber

7

“go-thurber”

Treccani

2

“go-treccani”

Trefethen

2

“go-trefethen”

Trid

6

“go-trid”

Trigonometric01

2

“go-trigonometric01”

Trigonometric02

2

“go-trigonometric02”

Tripod

2

“go-tripod”

Ursem01

2

“go-ursem01”

Ursem03

2

“go-ursem03”

Ursem04

2

“go-ursem04”

UrsemWaves

2

“go-ursemwaves”

VenterSobiezcczanskiSobieski

2

“go-ventersobiezcczanskisobieski”

Vincent

2

“go-vincent”

Watson

6

“go-watson”

Wavy

2

“go-wavy”

WayburnSeader01

2

“go-wayburnseader01”

WayburnSeader02

2

“go-wayburnseader02”

Weierstrass

2

“go-weierstrass”

Whitley

2

“go-whitley”

Wolfe

3

“go-wolfe”

XinSheYang01

2

“go-xinsheyang01”

XinSheYang02

2

“go-xinsheyang02”

XinSheYang03

2

“go-xinsheyang03”

XinSheYang04

2

“go-xinsheyang04”

Xor

9

“go-xor”

YaoLiu04

2

“go-yaoliu04”

YaoLiu09

2

“go-yaoliu09”

Zacharov

2

“go-zacharov”

ZeroSum

2

“go-zerosum”

Zettl

2

“go-zettl”

Zimmerman

2

“go-zimmerman”

Zirilli

2

“go-zirilli”