Author: erans
Date: Tue Feb 21 15:54:44 2012
New Revision: 1291882
URL: http://svn.apache.org/viewvc?rev=1291882&view=rev
Log:
Fixed links and some formatting in user guide.
Modified:
commons/proper/math/trunk/src/site/xdoc/userguide/optimization.xml
Modified: commons/proper/math/trunk/src/site/xdoc/userguide/optimization.xml
URL:
http://svn.apache.org/viewvc/commons/proper/math/trunk/src/site/xdoc/userguide/optimization.xml?rev=1291882&r1=1291881&r2=1291882&view=diff
==============================================================================
--- commons/proper/math/trunk/src/site/xdoc/userguide/optimization.xml
(original)
+++ commons/proper/math/trunk/src/site/xdoc/userguide/optimization.xml Tue Feb
21 15:54:44 2012
@@ -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 <
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
</p>
<source>
- private double[][] jacobian(double[] variables)
- {
+ private double[][] jacobian(double[] variables) {
double[][] jacobian = new double[x.size()][3];
for (int i = 0; i < jacobian.length; ++i) {
jacobian[i][0] = x.get(i) * x.get(i);
@@ -416,8 +415,7 @@ parameter is an ArrayList containing the
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 th
</p>
<source>
-private static class QuadraticProblem implements
DifferentiableMultivariateVectorialFunction, Serializable {
+private static class QuadraticProblem
+ implements DifferentiableMultivariateVectorFunction, Serializable {
private static final long serialVersionUID = 7072187082052755854L;
private List<Double> x;
@@ -474,11 +473,9 @@ private static class QuadraticProblem im
this.y.add(y);
}
- public double[] calculateTarget()
- {
+ public double[] calculateTarget() {
double[] target = new double[y.size()];
- for (int i = 0; i < y.size(); i++)
- {
+ for (int i = 0; i < y.size(); i++) {
target[i] = y.get(i).doubleValue();
}
return target;
@@ -522,34 +519,30 @@ optimal set of quadratic curve fitting p
<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);
-
- LevenbergMarquardtOptimizer optimizer
- = new LevenbergMarquardtOptimizer();
-
- double[] weights =
- { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 };
-
- double[] initialSolution = {1, 1, 1};
-
- VectorialPointValuePair optimum =
- optimizer.optimize(
- 100,
- problem,
- problem.calculateTarget(),
- weights,
- initialSolution);
+ 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();
+
+ final double[] weights = { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 };
+
+ final double[] initialSolution = {1, 1, 1};
+
+ PointVectorValuePair optimum = optimizer.optimize(100,
+ problem,
+ problem.calculateTarget(),
+ weights,
+ initialSolution);
- double[] optimalValues = optimum.getPoint();
+ final double[] optimalValues = optimum.getPoint();
System.out.println("A: " + optimalValues[0]);
System.out.println("B: " + 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-Ribière
+
href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateFunction.html">
+ DifferentiableMultivariateFunction</a>. Both the Fletcher-Reeves and
the Polak-Ribière
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>
