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An Open Source Java Framework for Multiobjective Optimization
<?xml version="1.0"?>
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<page>
<title>Examples using the MOEA Framework</title>
<description>Walk through introductory examples using the MOEA Framework</description>
<header>
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<content>
<h2>Examples</h2>
<p>
The following examples demonstrate the basic functionality provided by the
MOEA Framework. Links to the full source code are provided alongside each
code snippet. You may also find these and more examples in the demo
application on the <a href="downloads.html">downloads</a> page.
</p>
<ul>
<li><a href="#setup">Setup</a></li>
<li><a href="#example1">Example 1: Simple Run</a></li>
<li><a href="#example2">Example 2: Statistical Comparison of Algorithms</a></li>
<li><a href="#example3">Example 3: Collecting Runtime Dynamics</a></li>
<li><a href="#example4">Example 4: Defining New Problems</a></li>
<li><a href="#example5">Example 5: Defining Problems in C/C++</a></li>
</ul>
<div class="section">
<a name="setup" />
<h3>Setup</h3>
<p>
In order to run these examples or use the MOEA Framework, Java 6 (or a later
version) must be installed on your computer. The Java 6 development kit (JDK)
for Windows and Linux can be downloaded
<a href="http://www.oracle.com/technetwork/java/javase/downloads/jdk6u37-downloads-1859587.html">here</a>.
</p>
<p>
To run these examples, first download and extract the latest compiled
binaries from the <a href="downloads.html">downloads</a> page. Windows users
may extract the downloaded file using
<a href="http://www.7-zip.org/">7-zip</a>. The files will extract to a
folder called MOEAFramework-%VERSION%. This folder will look similar to:
</p>
<div class="files">
<ul>
<li class="folder">MOEAFramework-%VERSION%/</li>
<li class="empty">
<ul>
<li class="folder">examples/</li>
<li class="folder">javadoc/</li>
<li class="folder">lib/</li>
<li class="folder">licenses/</li>
<li class="folder">pf/</li>
<li>global.properties</li>
<li>HELP</li>
<li>launch-diagnostic-tool.bat</li>
<li>LICENSE</li>
<li>NEWS</li>
<li>README</li>
</ul>
</li>
</ul>
</div>
<p>
All of the examples below are in the examples/ folder. You may compile and
run an example using the following commands. Run these commands in the
Command Prompt from the MOEAFramework-%VERSION% folder.
</p>
<pre>
javac -cp "examples;lib/*" examples/Example1.java
java -cp "examples;lib/*" Example1</pre>
<p>
If you receive the message <i>'javac' is not recognized as an internal or
external command, operable program or batch file</i>, try the following steps
to setup your environment on
<a href="http://vietpad.sourceforge.net/javaonwindows.html">Windows</a> or
<a href="http://vietpad.sourceforge.net/javaonlinux.html">Linux</a>.
Unix/Linux users should replace the semicolons (;) with colons (:).
</p>
</div>
<div class="section">
<a name="example1" />
<h3>Example 1: Simple Run</h3>
<div class="files">
<ul>
<li><a href="Example1.java">Example1.java</a></li>
</ul>
</div>
<p>
The Executor provides all the necessary features to execute an algorithm on
a specific problem. For example, the following runs NSGA-II on the UF1 test
problem. The executor provides many useful functions, such as enabling
distributed evaluations across multiple cores or computers, checkpoints,
instrumentation, etc.
</p>
<div class="code">
<pre class="brush: java; toolbar: false;">
NondominatedPopulation result = new Executor()
.withProblem("UF1")
.withAlgorithm("NSGAII")
.withMaxEvaluations(10000)
.run();
</pre>
</div>
<p>
Since UF1 is a bi-objective problem, printing the results will list the two
objective function values for each non-dominated (Pareto optimal) solution
found by NSGA-II:
</p>
<div class="code">
<pre class="brush: plain; toolbar: false;">
Objective1 Objective2
0.9465 0.0466
0.1355 2.0513
0.1403 1.9960
0.2520 0.5120
...
</pre>
</div>
</div>
<div class="section">
<a name="example2" />
<h3>Example 2: Statistical Comparison of Algorithms</h3>
<div class="files">
<ul>
<li><a href="Example2.java">Example2.java</a></li>
</ul>
</div>
<p>
Statistical analyses are provided by the Analyzer. The Analyzer can display
the min, median, max and aggregate values for multiple performance indicators,
including hypervolume, generational distance, inverted generational distance,
additive ε-indicator, spacing and contribution. Additionally,
Kruskal-Wallis and Mann-Whitney U tests provide statistical significance
results. In the example below, we compare the algorithms using the
hypervolume metric.
</p>
<div class="code">
<pre class="brush: java; toolbar: false;">
<![CDATA[
String problem = "UF1";
String[] algorithms = { "NSGAII", "GDE3", "eMOEA" };
//setup the experiment
Executor executor = new Executor()
.withProblem(problem)
.withMaxEvaluations(10000);
Analyzer analyzer = new Analyzer()
.withProblem(problem)
.includeHypervolume()
.showStatisticalSignificance();
//run each algorithm for 50 seeds
for (String algorithm : algorithms) {
analyzer.addAll(algorithm,
executor.withAlgorithm(algorithm).runSeeds(50));
}
//print the results
analyzer.printAnalysis();
]]>
</pre>
</div>
<p>
Running this script produces the output shown below. We can see that GDE3 and
NSGA-II produce the best (largest) hypervolume values. Furthermore, we have
determined statistically that there is no significant difference in
performance between GDE3 and NSGA-II.
</p>
<div class="code">
<pre class="brush: plain; toolbar: false;">
GDE3:
Hypervolume:
Min: 0.4389785065649592
Median: 0.4974186560778636
Max: 0.535166930530847
Count: 50
Indifferent: [NSGAII]
eMOEA:
Hypervolume:
Min: 0.35003766343295073
Median: 0.47633216464734596
Max: 0.53311360537305
Count: 50
Indifferent: []
NSGAII:
Hypervolume:
Min: 0.38868701091987184
Median: 0.5040946740799506
Max: 0.5371138081508796
Count: 50
Indifferent: [GDE3]
</pre>
</div>
</div>
<div class="section">
<a name="example3" />
<h3>Example 3: Collecting Runtime Dynamics</h3>
<div class="files">
<ul>
<li><a href="Example3.java">Example3.java</a></li>
</ul>
</div>
<p>
Runtime dynamics provide insight into the behavior of an optimization
algorithm throughout a run. For instance, one can observe how solution
quality changes with the number of function evaluations (NFE). The
Instrumenter class records the runtime dynamics.
</p>
<div class="code">
<pre class="brush: java; toolbar: false;">
<![CDATA[
Instrumenter instrumenter = new Instrumenter()
.withProblem("UF1")
.withFrequency(100)
.attachElapsedTimeCollector()
.attachGenerationalDistanceCollector();
new Executor()
.withProblem("UF1")
.withAlgorithm("NSGAII")
.withMaxEvaluations(10000)
.withInstrumenter(instrumenter)
.run();
Accumulator accumulator = instrumenter.getLastAccumulator();
for (int i=0; i<accumulator.size("NFE"); i++) {
System.out.println(accumulator.get("NFE", i) + "\t" +
accumulator.get("Elapsed Time", i) + "\t" +
accumulator.get("GenerationalDistance", i));
}
]]>
</pre>
</div>
<p>
The output from this script, shown below, shows how the generational
distance metric changes over time. We see that NSGA-II is rapidly converging
to the reference set (the optimal solutions) since its generational
distance is converging to 0.
</p>
<div class="code">
<pre class="brush: plain; toolbar: false">
NFE Time Generational Distance
100 0.0256 0.7616
200 0.0421 0.6645
300 0.0543 0.4847
400 0.0636 0.4425
500 0.0713 0.4178
...
</pre>
</div>
</div>
<div class="section">
<a name="example4" />
<h3>Example 4: Defining New Problems</h3>
<div class="files">
<ul>
<li><a href="Example4.java">Example4.java</a></li>
</ul>
</div>
<p>
A number of methods are available to provide custom, user-defined problems
that cleanly integrate with all other components of the MOEA Framework. The
following demonstrates the two-objective DTLZ2 problem. Note that you only
need to define two methods: newSolution and evaluate. The newSolution
method defines the problem representation (the number and types of its
decision variables). The evaluate method takes a solution and computes its
objective function values.
</p>
<div class="code">
<pre class="brush: java; toolbar: false;">
<![CDATA[
public class DTLZ2 extends AbstractProblem {
public DTLZ2() {
super(11, 2);
}
public Solution newSolution() {
Solution solution = new Solution(getNumberOfVariables(),
getNumberOfObjectives());
for (int i = 0; i < getNumberOfVariables(); i++) {
solution.setVariable(i, new RealVariable(0.0, 1.0));
}
return solution;
}
public void evaluate(Solution solution) {
double[] x = CoreUtils.castVariablesToDoubleArray(solution);
double[] f = new double[numberOfObjectives];
int k = numberOfVariables - numberOfObjectives + 1;
double g = 0.0;
for (int i = numberOfVariables - k; i < numberOfVariables; i++) {
g += Math.pow(x[i] - 0.5, 2.0);
}
for (int i = 0; i < numberOfObjectives; i++) {
f[i] = 1.0 + g;
for (int j = 0; j < numberOfObjectives - i - 1; j++) {
f[i] *= Math.cos(0.5 * Math.PI * x[j]);
}
if (i != 0) {
f[i] *= Math.sin(0.5 * Math.PI * x[numberOfObjectives - i - 1]);
}
}
solution.setObjectives(f);
}
}
]]>
</pre>
</div>
<p>
This can subsequently be used with the Executor.
</p>
<div class="code">
<pre class="brush: java; toolbar: false;">
<![CDATA[
NondominatedPopulation result = new Executor()
.withProblemClass(DTLZ2.class)
.withAlgorithm("GDE3")
.withMaxEvaluations(10000)
.distributeOnAllCores()
.run();
]]>
</pre>
</div>
</div>
<div class="section">
<a name="example5" />
<h3>Example 5: Defining Problems in C/C++</h3>
<div class="files">
<ul>
<li><a href="Example5.java">Example5.java</a></li>
<li><a href="dtlz2.c">dtlz2.c</a></li>
</ul>
</div>
<p>
The MOEA Framework also supports defining new problems in other programming
languages, such as C/C++. First, a Java stub for the problem must be
created, as shown below. Note how the C/C++ executable is defined in the
constructor.
</p>
<div class="code">
<pre class="brush: java; toolbar: false;">
<![CDATA[
public static class DTLZ2 extends ExternalProblem {
public DTLZ2() throws IOException {
super("./auxiliary/c/dtlz2_stdio.exe");
}
@Override
public String getName() {
return "DTLZ2";
}
@Override
public int getNumberOfVariables() {
return 11;
}
@Override
public int getNumberOfObjectives() {
return 2;
}
@Override
public int getNumberOfConstraints() {
return 0;
}
@Override
public Solution newSolution() {
Solution solution = new Solution(getNumberOfVariables(),
getNumberOfObjectives());
for (int i = 0; i < getNumberOfVariables(); i++) {
solution.setVariable(i, new RealVariable(0.0, 1.0));
}
return solution;
}
}
]]>
</pre>
</div>
<p>
Next, we create the C/C++ program that defines the problem. Note the use of
several methods with names beginning with the MOEA_ prefix. These methods
are provided by the <code>moeaframework.h</code> library, which is
provided provided in the examples/ folder.
</p>
<div class="code">
<pre class="brush: cpp; toolbar: false;">
<![CDATA[
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include "moeaframework.h"
#define PI 3.14159265358979323846
int nvars = 11;
int nobjs = 2;
void evaluate(double* vars, double* objs) {
int i;
int j;
int k = nvars - nobjs + 1;
double g = 0.0;
for (i=nvars-k; i<nvars; i++) {
g += pow(vars[i] - 0.5, 2.0);
}
for (i=0; i<nobjs; i++) {
objs[i] = 1.0 + g;
for (j=0; j<nobjs-i-1; j++) {
objs[i] *= cos(0.5*PI*vars[j]);
}
if (i != 0) {
objs[i] *= sin(0.5*PI*vars[nobjs-i-1]);
}
}
}
int main(int argc, char* argv[]) {
double vars[nvars];
double objs[nobjs];
MOEA_Init(nobjs, 0);
while (MOEA_Next_solution() == MOEA_SUCCESS) {
MOEA_Read_doubles(nvars, vars);
evaluate(vars, objs);
MOEA_Write(objs, NULL);
}
MOEA_Terminate();
return EXIT_SUCCESS;
}
]]>
</pre>
</div>
<p>
The MOEA_Init method establishes a communication channel with the MOEA
Framework. This communication channel can use standard I/O streams or
sockets. The MOEA_Next_solution and MOEA_Read_doubles methods are used to
read the next solution to be evaluated. Once the solution is evaluated, the
computed objective function values are sent back to the MOEA Framework.
Finally, once all solutions are evaluated, we shutdown the communication
channel by calling MOEA_Terminate.
</p>
</div>
<div class="section">
<h3>Concluding Remarks</h3>
<p>
We hope that you find the MOEA Framework useful. We strive to make this
framework reliable and easy-to-use, and feedback from users like yourself
help us meet these goals. If you encounter any issues using this software,
please <a href="support.html">notify us</a>.
</p>
</div>
</content>
</page>