|
| [[Griewank function]]
| [[File:Griewank 2D Contour.svg|200px|Griewank'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(0, \dots, 0) = 0</math>
|<math>-\infty \le x_{i} \le \infty</math>, <math>1 \le i \le n</math>
| [[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>
|
|-
! 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>
|| <math>f(2.0052938,1.1944509) = -2.0239884</math>
|| <math>-2.25\le x \le 2.25</math>, <math>-2.5\le y \le 1.75</math>
|-
| 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-Levi contour.svg|200px|Gomez and Levy Function]]
|| <math>f(x,y) = 4 x^2 - 2.1 x^4 + \frac 1 3 x^6 + xy - 4y^2 +4 y^4 </math>,
subjected to:<math> -\sin(4 \pi x) + 2\sin^2(2 \pi y) \le 1.5 </math>
|| <math>f(0.08984201,-0.7126564) = -1.031628453</math>
|| <math>-1\le x \le 0.75</math>, <math>-1\le y \le 1</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>
|-
| 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-Levi contour.svg|200px|Gomez and Levy Function]]
|| <math>f(x,y) = 4 x^2 - 2.1 x^4 + \frac 1 3 x^6 + xy - 4y^2 +4 y^4 </math>,
subjected to:<math> -\sin(4 \pi x) + 2\sin^2(2 \pi y) \le 1.5 </math>
|| <math>f(0.08984201,-0.7126564) = -1.031628453</math>
|| <math>-1\le x \le 0.75</math>, <math>-1\le y \le 1</math>
|-
| '''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>
|| [[File:Estimation of Distribution Algorithm animation.gif|200px|Keane's bump function]]
||
|| <math>f(\boldsymbol{x}) = -\left| \frac{\left[ \sum_{i=1}^m \cos^4 (x_i) - 2 \prod_{i=1}^m \cos^2 (x_i) \right]}{{\left( \sum_{i=1}^m ix^2_i \right)}^{0.5}} \right| </math>,
subjected to: <math> 0.75 - \prod_{i=1}^m x_i < 0 </math>, and
<math> \sum_{i=1}^m x_i - 7.5m < 0 </math>
| 