Test functions for optimization: Difference between revisions

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{{Short description|Functions used to evaluate optimization algorithms}}

In applied mathematics, '''test functions''', known as '''artificial landscapes''', are useful to evaluate characteristics of optimization algorithms, such as: [[Rate of convergence|convergence rate]], precision, robustness and general performance.

* Convergence rate.

* Precision.

* Robustness.

* General performance.

Here some test functions are presented with the aim of giving an idea about the different situations that optimization algorithms have to face when coping with these kinds of problems. In the first part, some objective functions for single-objective optimization cases are presented. In the second part, test functions with their respective [[Pareto front|Pareto fronts]] for [[multi-objective optimization]] problems (MOP) are given.

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==Test functions for single-objective optimization==

{| class="sortable wikitable" style="text-align:center"

! Name

|-

! Name !! Plot !
! Formula !
! Global minimum !
! Search ___domain

|-

| [[Rastrigin function]]

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| <math>f(1,1) = 0</math>

| <math>-10\le x,y \le 10</math>

|-

*| [[ShekelGriewank function]]

|| [[File:SimionescuGriewank 2D contourContour.svg|200px|SimionescuGriewank's function]]

| <math>f(\boldsymbol{x})= 1+ \frac {1}{4000} \sum _{i=1}^n x_i^2 -\prod _{i=1}^n P_i(x_i)</math>, where <math>P_i(x_i)=\cos \left( \frac {x_i}{\sqrt {i}} \right)</math>

|| <math>f(1.0,1. \dots, 0) = 0</math>

|| <math>-1.5\infty \le xx_{i} \le 1.5\infty</math>, <math>-0.51 \le yi \le 2.5n</math>

|-

| [[Himmelblau's function]]

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|-

| Three-hump camel function

| [[File:Three-hump-camel contour.svg|200px|Three Hump Camel function]]

| <math>f(x,y) = 2x^{2} - 1.05x^{4} + \frac{x^{6}}{6} + xy + y^{2}</math>

| <math>f(0,0) = 0</math>

| <math>-5\le x,y \le 5</math>

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|-

| Cross-in-tray function

| [[File:Cross-in-tray contour.svg|200px|Cross-in-tray function]]

| <math>f(x,y) = -0.0001 \left[ \left| \sin x \sin y \exp \left(\left|100 - \frac{\sqrt{x^{2} + y^{2}}}{\pi} \right|\right)\right| + 1 \right]^{0.1}</math>

| <math>\text{Min} =

\begin{cases}

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| <math>-39.16617n < f(\underbrace{-2.903534, \ldots, -2.903534}_{n \text{ times}} ) < -39.16616n</math>

| <math>-5\le x_{i} \le 5</math>, <math>1\le i \le n</math>..

|

|-

| [[Shekel function]]

| [[Image:Shekel_2D.jpg|200px|A Shekel function in 2 dimensions and with 10 maxima]]

| <math>

f(\vecboldsymbol{x}) = \sum_{i = 1}^{m} \; \left( c_{i} + \sum\limits_{j = 1}^{n} (x_{j} - a_{ji})^2 \right)^{-1}

</math>

or, similarly,

<math>

f(x_1,x_2,...,x_{n-1},x_n) = \sum_{i = 1}^{m} \; \left( c_{i} + \sum\limits_{j = 1}^{n} (x_{j} - a_{ij})^2 \right)^{-1}

</math>

|

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|-

! Name !! Plot !! Formula !! Global minimum !! Search ___domain

|-

| Rosenbrock function constrained with a cubic and a line<ref>{{cite conference |author1=Simionescu, P.A. |author2=Beale, D. |title=New Concepts in Graphic Visualization of Objective Functions |conference=ASME 2002 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference |___location=Montreal, Canada |date=September 29 – October 2, 2002|pages=891–897 |url=http://faculty.tamucc.edu/psimionescu/PDFs/DETC02-DAC-34129.pdf |access-date=7 January 2017 }}</ref>

|| [[File:Rosenbrock cubic constraint.svg|200px|Rosenbrock function constrained with a cubic and a line]]

|| <math>f(x,y) = (1-x)^2 + 100(y-x^2)^2</math>,

subjected to: <math> (x-1)^3 - y + 1 \le 0 \text{ and } x + y - 2 \le 0 </math>

|| <math>f(1.0,1.0) = 0</math>

|| <math>-1.5\le x \le 1.5</math>, <math>-0.5\le y \le 2.5</math>

|-

| Rosenbrock function constrained to a disk<ref>{{Cite web|url=https://www.mathworks.com/help/optim/ug/example-nonlinear-constrained-minimization.html?requestedDomain=www.mathworks.com|title=Solve a Constrained Nonlinear Problem - MATLAB & Simulink|website=www.mathworks.com|access-date=2017-08-29}}</ref>

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|-

| '''Keane's bump function'''{{anchor|Keane's bump function}}<ref>{{cite journal |last1=Mishra |first1=Sudhanshu |title=Minimization of Keane’s Bump Function by the Repulsive Particle Swarm and the Differential Evolution Methods |date=5 May 2007 |url=https://econpapers.repec.org/paper/pramprapa/3098.htm |journal=MPRA Paper|publisher=University Library of Munich, Germany}}</ref>

| Gomez and Levy function (modified)<ref>{{cite journal |last1=Simionescu |first1=P.A. |date=2020 |title=A collection of bivariate nonlinear optimisation test problems with graphical representations |url= |journal=International Journal of Mathematical Modelling and Numerical Optimisation |volume=10 |issue=4 |pages=365–398 |doi=10.1504/IJMMNO.2020.110704 |access-date=}}</ref>

|| [[File:Gomez-LeviEstimation contourof Distribution Algorithm animation.svggif|200px|Gomez andKeane's Levybump Functionfunction]]

|| <math>f(\boldsymbol{x,y}) = 4 x^2 -\left| 2.\frac{\left[ \sum_{i=1}^m x\cos^4 +(x_i) - \frac2 \prod_{i=1}^m 3 x\cos^62 +(x_i) xy\right]}{{\left( -\sum_{i=1}^m 4yix^22_i +4 y\right)}^4{0.5}} \right| </math>,

subjected to: <math> 0.75 -\sin(4 \pi x) + 2\sinprod_{i=1}^2(2m \pix_i y)< \le 1.50 </math>, and

|| <math>f(0.08984201,-0.7126564) \sum_{i=1}^m x_i -1 7.0316284535m < 0 </math>

|| <math>-f((1\le.60025376,0.468675907)) x \le= -0.75364979746</math>, <math>-1\le y \le 1</math>

|| <math>f(x,y)0 =< (1-x)^2x_i +< 100(y-x^2)^210</math>,

|-

| [[Simionescu function]]<ref>{{cite book|last=Simionescu|first=P.A.|title=Computer Aided Graphing and Simulation Tools for AutoCAD Users|year=2014|publisher=CRC Press|___location=Boca Raton, FL|isbn=978-1-4822-5290-3|edition=1st}}</ref>

|| [[File:Simionescu contour.svg|200px|Simionescu function]]

|| <math>f(x,y) = 0.1xy</math>,

subjected to: <math> x^2+y^2\le\left[r_{T}+r_{S}\cos\left(n \arctan \frac{x}{y} \right)\right]^2</math>

<math>\text{where: } r_{T}=1, r_{S}=0.2 \text{ and } n = 8</math>

|| <math>f(\pm 0.84852813,\mp 0.84852813) = -0.072</math>

|| <math>-1.25\le x,y \le 1.25</math>

|}

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||<math>-3\le x,y \le 3</math>.

|}

== See also ==

{{commons category|Test functions (mathematical optimization)}}

* [[Shekel function]]

* [[Binh function]]

==References==

<references/>

== External links ==

<references/>

* [https://github.com/nathanrooy/landscapes landscapes]

{{DEFAULTSORT:Test functions for optimization}}

[[Category:Constraint programming]]

[[Category:Convex optimization]]

[[Category:Types ofTest functions for optimization| ]]

[[Category:Test items]]

[[Category:Mathematical optimization]]