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==Module Description==  
 
==Module Description==  
  
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=== Computation of the Objective ===  
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=== Multi-Objective Sphere ===  
  
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The single objective function ''f'' is a quadratic function. More precisely, ''f'' is computed as the square of all continuous and discrete variables.
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The multi-objective sphere function '''f''' is a set of quadratic functions. More precisely, each ''f<sub>i'' is computed as the square of all continuous variables.
  
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''f''('''x''') = sum<sub>i</sub>(x<sub>i</sub><sup>2</sup>)
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''f<sub>i</sub>''('''x''') = (''x<sub>i</sub>'' - 1)<sup>2</sup> + sum<sub>''j, j~=i''</sub> (x<sub>''j''</sub><sup>2</sup>)
  
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This function is also referred to as the ''sphere function''.
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For example
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''f<sub>1</sub>''('''x''') = (''x<sub>1</sub>'' - 1)<sup>2</sup> + ''x''<sub>2</sub><sup>2</sup>      + ''x''<sub>3</sub><sup>2 + ...
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''f<sub>2</sub>''('''x''') =  ''x<sub>1</sub>''<sup>2</sup>      + (''x<sub>2</sub>'' - 1)<sup>2</sup> + ''x''<sub>3</sub><sup>2 + ...
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...
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The Pareto front for the two-objective case is given in the following picture:
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{|
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| [[Image:moo_sphere_ds.png|100x100px|center|thumb|300x300px|Design space]]
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| [[Image:moo_sphere.png|100x100px|center|thumb|300x300px|Objective space]]
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|}
  
 
==Usage==
 
==Usage==

Aktuelle Version vom 7. April 2019, 17:29 Uhr

Summary

The module ContinuousMOTestProblems contains a set of multi-objective optimization problems, which can be used to test optimization algorithms.

Properties

General

Algorithm deterministic test function
Design Variables continuous design variables only
Objectives multi-objective for minimization
Constraints none
Boundaries problem dependent
Initial Search Region problem dependent
Typical X not set

Connections

Starting at his module none
Ending at this module Connections of type optimization

Actions

Name Description
- -

Options

The options are the number of discrete and continuous variables and an optional time delay for per evaluation.

Module Description

Multi-Objective Sphere

The multi-objective sphere function f is a set of quadratic functions. More precisely, each fi is computed as the square of all continuous variables.

fi(x) = (xi - 1)2 + sumj, j~=i (xj2)

For example

f1(x) = (x1 - 1)2 + x22 + x32 + ...

f2(x) = x12 + (x2 - 1)2 + x32 + ...

...

The Pareto front for the two-objective case is given in the following picture:

Design space
Objective space

Usage

-

Source Code

https://sourceforge.net/p/opendino/code/HEAD/tree/trunk/src/org/opendino/modules/problems/ContinuousMOTestProblems.java

References

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Abgerufen von „http://opendino.org/wiki/index.php?title=Documentation/Modules/ContinuousMOTestProblems&oldid=256“