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经典测试函数.pdf
Contents
1 Introduction
1.1 Examples of Parametric Optimization
1.2 Examples of Multi-objective Optimization
2 Parametric Optimization
2.1 De Jong's function 1
2.2 Axis parallel hyper-ellipsoid function
2.3 Rotated hyper-ellipsoid function
2.4 Moved axis parallel hyper-ellipsoid function
2.5 Rosenbrock's valley (De Jong's function 2)
2.6 Rastrigin's function 6
2.7 Schwefel's function 7
2.8 Griewangk's function 8
2.9 Sum of different power function 9
2.10 Ackley's Path function 10
2.11 Langermann's function 11
2.12 Michalewicz's function 12
2.13 Branins's rcos function
2.14 Easom's function
2.15 Goldstein-Price's function
2.16 Six-hump camel back function
3 Multi-objective Optimization
3.1 Fonseca's function 1 and 2
Index
GEATbx com
Genetic and Evolutionary Algorithm Toolbox for Matlab
GEATbx Examples
Examples of
Objective Functions
by:
Hartmut Pohlheim
GEATbx version 3.8
(December 2006)
www.geatbx.com
support@geatbx.com
Contents
1 Introduction ..................................................................................................... 1
1.1 Examples of Parametric Optimization............................................................................ 1
1.2 Examples of Multi-objective Optimization.................................................................... 1
2 Parametric Optimization................................................................................ 3
2.1 De Jong's function 1 ....................................................................................................... 3
2.2 Axis parallel hyper-ellipsoid function............................................................................ 3
2.3 Rotated hyper-ellipsoid function .................................................................................... 4
2.4 Moved axis parallel hyper-ellipsoid function................................................................. 5
2.5 Rosenbrock's valley (De Jong's function 2) ................................................................... 5
2.6 Rastrigin's function 6...................................................................................................... 6
2.7 Schwefel's function 7...................................................................................................... 7
2.8 Griewangk's function 8................................................................................................... 7
2.9 Sum of different power function 9 ................................................................................. 8
2.10 Ackley's Path function 10............................................................................................. 9
2.11 Langermann's function 11 .......................................................................................... 10
2.12 Michalewicz's function 12.......................................................................................... 10
2.13 Branins's rcos function ............................................................................................... 11
2.14 Easom's function......................................................................................................... 12
2.15 Goldstein-Price's function .......................................................................................... 12
2.16 Six-hump camel back function................................................................................... 13
3 Multi-objective Optimization....................................................................... 15
3.1 Fonseca's function 1 and 2............................................................................................ 15
Index ................................................................................................................... 17
www.geatbx.com
GEATbx Examples
1 Introduction
This document describes a number of test functions implemented for use with the Genetic and Evolutionary Al-
gorithm Toolbox for Matlab (GEATbx). These functions are drawn from the literature on evolutionary algo-
rithms and global optimization. The first Section describes a set of common parametric test problems imple-
mented as Matlab m-files. The second Section presents a number of dynamic systems, implemented in Simulink,
as s-files and m-files as appropriate.
1.1 Examples of Parametric Optimization
Each of the functions in Chapter 2 is described by the function definition, one or more 3-D graphics to show the
properties of the function and a description of the features of the function.
1.2 Examples of Multi-objective Optimization
The functions in Chapter 3 constitute multi-objective example functions. For each of them the definition and a
description of the features of the function are given. Plots of the PARETO-front in search and solution space
enhance the understanding of the functions. If useful, 3-D graphics showing the search space are provided.
All of the test function implementations are scaleable, i.e. the functions can be called with as many dimensions
as necessary and the default dimension of the test functions is adjustable via a single parameter value inside the
function.
For writing own objective functions see Writing objective functions.
www.geatbx.com
GEATbx Examples
2 Parametric Optimization
2.1 De Jong's function 1
The simplest test function is De Jong's function 1. It is also known as sphere model. It is continuos, convex and
unimodal.
function definition:
( )
xf
1
n
= ∑
x
i
x
2
i
≤
12.5
−
12.5
≤
i
1
=
f1(x)=sum(x(i)^2), i=1:n, -5.12<=x(i)<=5.12.
global minimum:
f(x)=0, x(i)=0, i=1:n.
This function is implemented in objfun1.
Fig. 2-1:
Visualization of De Jong's function 1 using different domains of the variables; however, both
graphics look similar, just the scaling changed; left: surf plot of the function in a very large
area from -500 to 500 for each of both variables, right: the function at a smaller area from -10
to 10
2.2 Axis parallel hyper-ellipsoid function
The axis parallel hyper-ellipsoid is similar to De Jong's function 1. It is also known as the weighted sphere
model. Again, it is continuos, convex and unimodal.
function definition:
( )
x
n
= ∑
f
a1
xi
2
⋅
i
−
12.5
≤
f1a(x)=sum(i·x(i)^2), i=1:n, -5.12<=x(i)<=5.12.
global minimum:
1
=
i
≤
12.5
x
i
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GEATbx Examples
4
2 Parametric Optimization
f(x)=0; x(i)= 0, i=1:n.
This function is implemented in objfun1a.
Fig. 2-2:
Visualization of Axis parallel hyper-ellipsoid function; surf/mesh plot of the function in an
area from -5 to 5
2.3 Rotated hyper-ellipsoid function
An extension of the axis parallel hyper-ellipsoid is Schwefel's function1.2. With respect to the coordinate axes,
this function produces rotated hyper-ellipsoids. It is continuos, convex and unimodal.
function definition:
f
b1
( )
x
= ∑ ∑
x
j
i
1
=
j
1
=
2
−
536.65
≤
x
i
≤
536.65
n
i
⎛
⎜⎜
⎝
⎞
⎟⎟
⎠
f1b(x)=sum(sum(x(j)^2), j=1:i), i=1:n, -65.536<=x(i)<=65.536.
global minimum:
f(x)=0; x(i)=0, i=1:n.
This function is implemented in objfun1b.
Fig. 2-3:
Visualization of Rotated hyper-ellipsoid function; surf/mesh plot of the first two variables in
an area from –50 to 50
GEATbx Examples
www.geatbx.com
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