Test functions for optimization: Difference between revisions

Revision as of 17:32, 5 February 2025

Latest revision as of 11:53, 17 July 2025

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Line 92:

| [[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>

Line 187:

| [[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>

|

Line 229 ⟶ 225:

| '''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>