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Fixed links and some formatting in user guide. 

git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@1291882 13f79535-47bb-0310-9956-ffa450edef68
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Gilles Sadowski 2012-02-21 15:54:44 +00:00
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commit 829bb2b714
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149
src/site/xdoc/userguide/optimization.xml
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@ -63,21 +63,21 @@
are only four interfaces defining the common behavior of optimizers, one for each
supported type of objective function:
<ul>
<li><a href="../apidocs/org/apache/commons/math3/optimization/UnivariateRealOptimizer.html">
UnivariateRealOptimizer</a> for <a
href="../apidocs/org/apache/commons/math3/analysis/UnivariateRealFunction.html">
<li><a href="../apidocs/org/apache/commons/math3/optimization/univariate/UnivariateOptimizer.html">
UnivariateOptimizer</a> for <a
href="../apidocs/org/apache/commons/math3/analysis/UnivariateFunction.html">
univariate real functions</a></li>
<li><a href="../apidocs/org/apache/commons/math3/optimization/MultivariateRealOptimizer.html">
MultivariateRealOptimizer</a> for <a
href="../apidocs/org/apache/commons/math3/analysis/MultivariateRealFunction.html">
<li><a href="../apidocs/org/apache/commons/math3/optimization/MultivariateOptimizer.html">
MultivariateOptimizer</a> for <a
href="../apidocs/org/apache/commons/math3/analysis/MultivariateFunction.html">
multivariate real functions</a></li>
<li><a href="../apidocs/org/apache/commons/math3/optimization/DifferentiableMultivariateRealOptimizer.html">
DifferentiableMultivariateRealOptimizer</a> for <a
href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateRealFunction.html">
<li><a href="../apidocs/org/apache/commons/math3/optimization/DifferentiableMultivariateOptimizer.html">
DifferentiableMultivariateOptimizer</a> for <a
href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateFunction.html">
differentiable multivariate real functions</a></li>
<li><a href="../apidocs/org/apache/commons/math3/optimization/DifferentiableMultivariateVectorialOptimizer.html">
DifferentiableMultivariateVectorialOptimizer</a> for <a
href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateVectorialFunction.html">
<li><a href="../apidocs/org/apache/commons/math3/optimization/DifferentiableMultivariateVectorOptimizer.html">
DifferentiableMultivariateVectorOptimizer</a> for <a
href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateVectorFunction.html">
differentiable multivariate vectorial functions</a></li>
</ul>
</p>
@ -85,15 +85,15 @@
<p>
Despite there are only four types of supported optimizers, it is possible to optimize
a transform a <a
href="../apidocs/org/apache/commons/math3/analysis/MultivariateVectorialFunction.html">
href="../apidocs/org/apache/commons/math3/analysis/MultivariateVectorFunction.html">
non-differentiable multivariate vectorial function</a> by converting it to a <a
href="../apidocs/org/apache/commons/math3/analysis/MultivariateRealFunction.html">
href="../apidocs/org/apache/commons/math3/analysis/MultivariateFunction.html">
non-differentiable multivariate real function</a> thanks to the <a
href="../apidocs/org/apache/commons/math3/optimization/LeastSquaresConverter.html">
LeastSquaresConverter</a> helper class. The transformed function can be optimized using
any implementation of the <a
href="../apidocs/org/apache/commons/math3/optimization/MultivariateRealOptimizer.html">
MultivariateRealOptimizer</a> interface.
href="../apidocs/org/apache/commons/math3/optimization/MultivariateOptimizer.html">
MultivariateOptimizer</a> interface.
</p>
<p>
@ -106,8 +106,8 @@
</subsection>
<subsection name="12.2 Univariate Functions" href="univariate">
<p>
A <a href="../apidocs/org/apache/commons/math3/optimization/UnivariateRealOptimizer.html">
UnivariateRealOptimizer</a> is used to find the minimal values of a univariate real-valued
A <a href="../apidocs/org/apache/commons/math3/optimization/univariate/UnivariateOptimizer.html">
UnivariateOptimizer</a> is used to find the minimal values of a univariate real-valued
function <code>f</code>.
</p>
<p>
@ -174,10 +174,10 @@
<p>
The first two simplex-based methods do not handle simple bounds constraints by themselves.
However there are two adapters(<a
href="../apidocs/org/apache/commons/math3/optimization/direct/MultivariateRealFunctionMappingAdapter.html">
MultivariateRealFunctionMappingAdapter</a> and <a
href="../apidocs/org/apache/commons/math3/optimization/direct/MultivariateRealFunctionPenaltyAdapter.html">
MultivariateRealFunctionPenaltyAdapter</a>) that can be used to wrap the user function in
href="../apidocs/org/apache/commons/math3/optimization/direct/MultivariateFunctionMappingAdapter.html">
MultivariateFunctionMappingAdapter</a> and <a
href="../apidocs/org/apache/commons/math3/optimization/direct/MultivariateFunctionPenaltyAdapter.html">
MultivariateFunctionPenaltyAdapter</a>) that can be used to wrap the user function in
such a way the wrapped function is unbounded and can be used with these optimizers, despite
the fact the underlying function is still bounded and will be called only with feasible
points that fulfill the constraints. Note however that using these adapters are only a
@ -238,8 +238,8 @@
<p>
In order to solve a vectorial optimization problem, the user must provide it as
an object implementing the <a
href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateVectorialFunction.html">
DifferentiableMultivariateVectorialFunction</a> interface. The object will be provided to
href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateVectorFunction.html">
DifferentiableMultivariateVectorFunction</a> interface. The object will be provided to
the <code>estimate</code> method of the optimizer, along with the target and weight arrays,
thus allowing the optimizer to compute the residuals at will. The last parameter to the
<code>estimate</code> method is the point from which the optimizer will start its
@ -251,9 +251,10 @@
<dd>
We are looking to find the best parameters [a, b, c] for the quadratic function <b><tt> f(x)=a*x^2 + b*x + c </tt></b>.
The data set below was generated using [a = 8, b = 10, c = 16]. A random number between zero and one was added
to each y value calculated.
We are looking to find the best parameters [a, b, c] for the quadratic function
<b><code>f(x) = a x<sup>2</sup> + b x + c</code></b>.
The data set below was generated using [a = 8, b = 10, c = 16].
A random number between zero and one was added to each y value calculated.
<table cellspacing="0" cellpadding="3">
<tr>
@ -303,7 +304,7 @@
</table>
<p>
First we need to implement the interface <a href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateVectorialFunction.html">DifferentiableMultivariateVectorialFunction</a>.
First we need to implement the interface <a href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateVectorFunction.html">DifferentiableMultivariateVectorFunction</a>.
This requires the implementation of the method signatures:
</p>
@ -318,24 +319,23 @@ We'll tackle the implementation of the <code>MultivariateMatrixFunction jacobian
In this case the Jacobian is the partial derivative of the function with respect
to the parameters a, b and c. These derivatives are computed as follows:
<ul>
<li>d(ax^2+bx+c)/da = x2</li>
<li>d(ax^2+bx+c)/db = x</li>
<li>d(ax^2+bx+c)/dc = 1</li>
<li>d(ax<sup>2</sup> + bx + c)/da = x<sup>2</sup></li>
<li>d(ax<sup>2</sup> + bx + c)/db = x</li>
<li>d(ax<sup>2</sup> + bx + c)/dc = 1</li>
</ul>
</p>
<p>
For a quadratic which has three variables the Jacobian Matrix will have three columns, one for each variable, and the number
of rows will equal the number of rows in our data set, which in this case is ten. So for example for <b><tt>[a = 1, b=1, c=1]</tt></b>
the Jacobian Matrix is (Exluding the first column which shows the value of x):
of rows will equal the number of rows in our data set, which in this case is ten. So for example for <tt>[a = 1, b = 1, c = 1]</tt>, the Jacobian Matrix is (excluding the first column which shows the value of x):
</p>
<table cellspacing="0" cellpadding="3">
<tr>
<td valign="bottom" align="left" style=" font-size:10pt;"><b>x</b></td>
<td valign="bottom" align="left" style=" font-size:10pt;"><b>d(ax^2+bx+c)/da</b></td>
<td valign="bottom" align="left" style=" font-size:10pt;"><b>d(ax^2+bx+c)/db</b></td>
<td valign="bottom" align="left" style=" font-size:10pt;"><b>d(ax^2+bx+c)/dc</b></td>
<td valign="bottom" align="left" style=" font-size:10pt;"><b>d(ax<sup>2</sup> + bx + c)/da</b></td>
<td valign="bottom" align="left" style=" font-size:10pt;"><b>d(ax<sup>2</sup> + bx + c)/db</b></td>
<td valign="bottom" align="left" style=" font-size:10pt;"><b>d(ax<sup>2</sup> + bx + c)/dc</b></td>
</tr>
<tr>
<td valign="bottom" align="center" style=" font-size:10pt;">1</td>
@ -405,8 +405,7 @@ parameter is an ArrayList containing the independent values of the data set):
</p>
<source>
private double[][] jacobian(double[] variables)
{
private double[][] jacobian(double[] variables) {
double[][] jacobian = new double[x.size()][3];
for (int i = 0; i &lt; jacobian.length; ++i) {
jacobian[i][0] = x.get(i) * x.get(i);
@ -416,8 +415,7 @@ parameter is an ArrayList containing the independent values of the data set):
return jacobian;
}
public MultivariateMatrixFunction jacobian()
{
public MultivariateMatrixFunction jacobian() {
return new MultivariateMatrixFunction() {
private static final long serialVersionUID = -8673650298627399464L;
public double[][] value(double[] point) {
@ -458,7 +456,8 @@ Below is the the class containing all the implementation details
</p>
<source>
private static class QuadraticProblem implements DifferentiableMultivariateVectorialFunction, Serializable {
private static class QuadraticProblem
implements DifferentiableMultivariateVectorFunction, Serializable {
private static final long serialVersionUID = 7072187082052755854L;
private List&lt;Double&gt; x;
@ -474,11 +473,9 @@ private static class QuadraticProblem implements DifferentiableMultivariateVecto
this.y.add(y);
}
public double[] calculateTarget()
{
public double[] calculateTarget() {
double[] target = new double[y.size()];
for (int i = 0; i &lt; y.size(); i++)
{
for (int i = 0; i &lt; y.size(); i++) {
target[i] = y.get(i).doubleValue();
}
return target;
@ -522,34 +519,30 @@ optimal set of quadratic curve fitting parameters:
<source>
QuadraticProblem problem = new QuadraticProblem();
problem.addPoint (1, 34.234064369);
problem.addPoint (2, 68.2681162306);
problem.addPoint (3, 118.6158990846);
problem.addPoint (4, 184.1381972386);
problem.addPoint (5, 266.5998779163);
problem.addPoint (6, 364.1477352516);
problem.addPoint (7, 478.0192260919);
problem.addPoint (8, 608.1409492707);
problem.addPoint (9, 754.5988686671);
problem.addPoint (10, 916.1288180859);
problem.addPoint(1, 34.234064369);
problem.addPoint(2, 68.2681162306);
problem.addPoint(3, 118.6158990846);
problem.addPoint(4, 184.1381972386);
problem.addPoint(5, 266.5998779163);
problem.addPoint(6, 364.1477352516);
problem.addPoint(7, 478.0192260919);
problem.addPoint(8, 608.1409492707);
problem.addPoint(9, 754.5988686671);
problem.addPoint(10, 916.1288180859);
LevenbergMarquardtOptimizer optimizer
= new LevenbergMarquardtOptimizer();
LevenbergMarquardtOptimizer optimizer = new LevenbergMarquardtOptimizer();
double[] weights =
{ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 };
final double[] weights = { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 };
double[] initialSolution = {1, 1, 1};
final double[] initialSolution = {1, 1, 1};
VectorialPointValuePair optimum =
optimizer.optimize(
100,
problem,
problem.calculateTarget(),
weights,
initialSolution);
PointVectorValuePair optimum = optimizer.optimize(100,
problem,
problem.calculateTarget(),
weights,
initialSolution);
double[] optimalValues = optimum.getPoint();
final double[] optimalValues = optimum.getPoint();
System.out.println(&quot;A: &quot; + optimalValues[0]);
System.out.println(&quot;B: &quot; + optimalValues[1]);
@ -574,14 +567,14 @@ C: 16.324008168386605
href="../apidocs/org/apache/commons/math3/optimization/general/NonLinearConjugateGradientOptimizer.html">
NonLinearConjugateGradientOptimizer</a> class provides a non-linear conjugate gradient algorithm
to optimize <a
href="../apidocs/org/apache/commons/math3/optimization/DifferentiableMultivariateRealFunction.html">
DifferentiableMultivariateRealFunction</a>. Both the Fletcher-Reeves and the Polak-Ribi&#232;re
href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateFunction.html">
DifferentiableMultivariateFunction</a>. Both the Fletcher-Reeves and the Polak-Ribi&#232;re
search direction update methods are supported. It is also possible to set up a preconditioner
or to change the line-search algorithm of the inner loop if desired (the default one is a Brent
solver).
</p>
<p>
The <a href="../apidocs/org/apache/commons/math3/optimization/general/PowellOptimizer.html">
The <a href="../apidocs/org/apache/commons/math3/optimization/direct/PowellOptimizer.html">
PowellOptimizer</a> provides an optimization method for non-differentiable functions.
</p>
</subsection>
@ -612,8 +605,8 @@ C: 16.324008168386605
CurveFitter</a> class provides curve fitting for general curves. Users must
provide their own implementation of the curve template as a class implementing
the <a
href="../apidocs/org/apache/commons/math3/optimization/fitting/ParametricRealFunction.html">
ParametricRealFunction</a> interface and they must provide the initial guess of the
href="../apidocs/org/apache/commons/math3/analysis/ParametricUnivariateFunction.html">
ParametricUnivariateFunction</a> interface and they must provide the initial guess of the
parameters. The more specialized <a
href="../apidocs/org/apache/commons/math3/optimization/fitting/PolynomialFitter.html">
PolynomialFitter</a> and <a
@ -625,11 +618,11 @@ C: 16.324008168386605
An example of fitting a polynomial is given here:
</p>
<source>PolynomialFitter fitter = new PolynomialFitter(degree, new LevenbergMarquardtOptimizer());
fitter.addObservedPoint(-1.00, 2.021170021833143);
fitter.addObservedPoint(-0.99 2.221135431136975);
fitter.addObservedPoint(-0.98 2.09985277659314);
fitter.addObservedPoint(-0.97 2.0211192647627025);
// lots of lines ommitted
fitter.addObservedPoint(-1.00, 2.021170021833143);
fitter.addObservedPoint(-0.99, 2.221135431136975);
fitter.addObservedPoint(-0.98, 2.09985277659314);
fitter.addObservedPoint(-0.97, 2.0211192647627025);
// ... Lots of lines omitted ...
fitter.addObservedPoint( 0.99, -2.4345814727089854);
PolynomialFunction fitted = fitter.fit();
</source>