Test problems: Difference between revisions
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==Test problems== | ==Test problems== | ||
=== | |||
===Ackley function=== | |||
[[File:BlueM.Opt Ackley function.png|thumb|right|Ackley function]] | |||
The Ackley function is a non-convex function used as a performance test problem for optimization algorithms. It was proposed by David Ackley in his 1987 PhD dissertation{{:Literature:Ackley 1987|}}. | |||
On a 2-dimensional domain it is defined by: | |||
: <math> | |||
\begin{align} | |||
f(x,y) = -20&{}\exp\left[-0.2\sqrt{0.5(x^2+y^2)}\,\right] \\ | |||
& {} -\exp\left[0.5\left(\cos 2\pi x + \cos 2\pi y \right)\right] + e + 20 | |||
\end{align} | |||
</math>{{:Literature:Bäck 1996|}} | |||
Its global optimum point is | |||
:<math>f(0,0) = 0.</math><br clear="all" /> | |||
===Beale problem=== | ===Beale problem=== | ||
[[File:Beale Sensiplot.png|thumb|Beale function (created with [[SensiPlot]])]] | [[File:Beale Sensiplot.png|thumb|right|Beale function (created with [[SensiPlot]])]] | ||
[[File:Beale_ani.gif|thumb| | [[File:Beale_ani.gif|thumb|right|Beale problem being solved with [[PES]] (animation)]] | ||
Finding the minimum of the Beale function{{:Literature:Beale 1958|}}. | Finding the minimum of the Beale function{{:Literature:Beale 1958|}}. | ||
* Parameters: 2 | * Parameters: 2 | ||
| Line 17: | Line 31: | ||
Global minimum: <code>f(3, 0.5) = 0</code><br clear="all" /> | Global minimum: <code>f(3, 0.5) = 0</code><br clear="all" /> | ||
===Box=== | |||
[[File:EVO Box screenshot.png|thumb|Box]] | |||
Multicriteria test problem (circle) with two contraints<br clear="all"/> | |||
===CONSTR=== | |||
[[File:CONSTR_ani.gif|thumb|CONSTR]] | |||
Multicriteria test problem (convex) with two constraints<br clear="all" /> | |||
===Deb 1=== | |||
Multicriteria test problem (convex) | |||
===Dependent parameters=== | |||
Parameter dependency: Y > X | |||
===Flood Mitigation=== | |||
[[File:TP FloodMitigation.png|thumb|Flood Mitigation]] | |||
Multicriteria Problem Flood Mitigation and Hydropower Generation<ref>'''Sharma, Ajay''' (2008): Inflow prediction and optimal operation of reservoir system during flood by the combined application of ANN and different Optimization techniques. Master Thesis, Institute of Hydraulic and Water Resources Engineering, Technische Universität Darmstadt.</ref><br clear="all"/> | |||
===Schwefel 2.4 problem=== | ===Schwefel 2.4 problem=== | ||
Find the minimum (xi=1, F(x)=0) | Find the minimum (xi=1, F(x)=0) | ||
=== | ===Sine function=== | ||
Fit parameters to the sine function | |||
===Zitzler/Deb T1=== | ===Zitzler/Deb T1=== | ||
| Line 31: | Line 63: | ||
===Zitzler/Deb T3=== | ===Zitzler/Deb T3=== | ||
[[File: | [[File:Zitzler deb t3 ani.gif|thumb|right|Zitzler/Deb T3 being solved with [[PES]] (animation)]] | ||
Multicriteria test problem (convex, non-continuous) | Multicriteria test problem (convex, non-continuous)<br clear="all" /> | ||
===Zitzler/Deb T4=== | ===Zitzler/Deb T4=== | ||
Multicriteria test problem (convex) | Multicriteria test problem (convex) | ||
==References== | ==References== | ||
Latest revision as of 01:18, 19 May 2023
BlueM.Opt | Download | Usage | Development
List of test problems available in BlueM.Opt. Some of them are taken from Moré et al. (1981)[1].
Test problems
Ackley function
The Ackley function is a non-convex function used as a performance test problem for optimization algorithms. It was proposed by David Ackley in his 1987 PhD dissertation[2].
On a 2-dimensional domain it is defined by:
- [math]\displaystyle{ \begin{align} f(x,y) = -20&{}\exp\left[-0.2\sqrt{0.5(x^2+y^2)}\,\right] \\ & {} -\exp\left[0.5\left(\cos 2\pi x + \cos 2\pi y \right)\right] + e + 20 \end{align} }[/math][3]
Its global optimum point is
- [math]\displaystyle{ f(0,0) = 0. }[/math]
Beale problem
Beale function (created with SensiPlot)
Beale problem being solved with PES (animation)
Finding the minimum of the Beale function[4].
- Parameters: 2
- Objective functions: 1
[math]\displaystyle{ f(x,y)=(1.5-x(1-y))^2+(2.25-x(1-y^2))^2+(2.625-x(1-y^3))^2 }[/math]
Global minimum: f(3, 0.5) = 0
Box
Multicriteria test problem (circle) with two contraints
CONSTR
Multicriteria test problem (convex) with two constraints
Deb 1
Multicriteria test problem (convex)
Dependent parameters
Parameter dependency: Y > X
Flood Mitigation
Multicriteria Problem Flood Mitigation and Hydropower Generation[5]
Schwefel 2.4 problem
Find the minimum (xi=1, F(x)=0)
Sine function
Fit parameters to the sine function
Zitzler/Deb T1
Multicriteria test problem (convex)
Zitzler/Deb T2
Multicriteria test problem (concave)
Zitzler/Deb T3
Zitzler/Deb T3 being solved with PES (animation)
Multicriteria test problem (convex, non-continuous)
Zitzler/Deb T4
Multicriteria test problem (convex)
References
- ↑ Moré, J.J., Garbow, B.S. and Hillstrom, K.E. (1981): Testing Unconstrained Optimization Software, ACM Transactions on Mathematical Software (TOMS) 7:1, p. 17-41, doi:10.1145/355934.355936
- ↑ Ackley, D. H. (1987): "A connectionist machine for genetic hillclimbing", Kluwer Academic Publishers, Boston MA.
- ↑ Bäck, Thomas (1996): "Artificial Landscapes". Evolutionary Algorithms in Theory and Practice. Oxford University Press. p. 142. doi:10.1093/oso/9780195099713.003.0008. ISBN 978-0-19-509971-3.
- ↑ Beale, E. M. L. (1958): On an iterative method of finding a local minimum of a function of more than one variable. Technical Report 25, Statistical Techniques Research Group, Princeton University.
- ↑ Sharma, Ajay (2008): Inflow prediction and optimal operation of reservoir system during flood by the combined application of ANN and different Optimization techniques. Master Thesis, Institute of Hydraulic and Water Resources Engineering, Technische Universität Darmstadt.