<td valign="bottom" align="center" style=" font-size:10pt;">Y </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">1 <td valign="bottom" align="center" style=" font-size:10pt;">34.234064369 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">2 <td valign="bottom" align="center" style=" font-size:10pt;">68.2681162306108 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">3 <td valign="bottom" align="center" style=" font-size:10pt;">118.615899084602 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">4 <td valign="bottom" align="center" style=" font-size:10pt;">184.138197238557 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">5 <td valign="bottom" align="center" style=" font-size:10pt;">266.599877916276 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">6 <td valign="bottom" align="center" style=" font-size:10pt;">364.147735251579 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">7 <td valign="bottom" align="center" style=" font-size:10pt;">478.019226091914 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">8 <td valign="bottom" align="center" style=" font-size:10pt;">608.140949270688 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">9 <td valign="bottom" align="center" style=" font-size:10pt;">754.598868667148 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">10 <td valign="bottom" align="center" style=" font-size:10pt;">916.128818085883 </tr> </table> <p> First we need to implement the interface <a href="../apidocs/org/apache/commons/math3/analysis/DifferentiableMultivariateVectorFunction.html">DifferentiableMultivariateVectorFunction. This requires the implementation of the method signatures: </p> <ul> <li>MultivariateMatrixFunction jacobian() <li>double[] value(double[] point) </ul> <p> We'll tackle the implementation of the <code>MultivariateMatrixFunction jacobian() method first. You may wish to familiarize yourself with what a Jacobian Matrix is. 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(ax² + bx + c)/da = x² <li>d(ax² + bx + c)/db = x <li>d(ax² + bx + c)/dc = 1 </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 <tt>[a = 1, b = 1, c = 1], 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;">x <td valign="bottom" align="left" style=" font-size:10pt;">d(ax² + bx + c)/da <td valign="bottom" align="left" style=" font-size:10pt;">d(ax² + bx + c)/db <td valign="bottom" align="left" style=" font-size:10pt;">d(ax² + bx + c)/dc </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">1 <td valign="bottom" align="center" style=" font-size:10pt;">1 <td valign="bottom" align="center" style=" font-size:10pt;">1 <td valign="bottom" align="center" style=" font-size:10pt;">1 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">2 <td valign="bottom" align="center" style=" font-size:10pt;">4 <td valign="bottom" align="center" style=" font-size:10pt;">2 <td valign="bottom" align="center" style=" font-size:10pt;">1 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">3 <td valign="bottom" align="center" style=" font-size:10pt;">9 <td valign="bottom" align="center" style=" font-size:10pt;">3 <td valign="bottom" align="center" style=" font-size:10pt;">1 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">4 <td valign="bottom" align="center" style=" font-size:10pt;">16 <td valign="bottom" align="center" style=" font-size:10pt;">4 <td valign="bottom" align="center" style=" font-size:10pt;">1 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">5 <td valign="bottom" align="center" style=" font-size:10pt;">25 <td valign="bottom" align="center" style=" font-size:10pt;">5 <td valign="bottom" align="center" style=" font-size:10pt;">1 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">6 <td valign="bottom" align="center" style=" font-size:10pt;">36 <td valign="bottom" align="center" style=" font-size:10pt;">6 <td valign="bottom" align="center" style=" font-size:10pt;">1 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">7 <td valign="bottom" align="center" style=" font-size:10pt;">49 <td valign="bottom" align="center" style=" font-size:10pt;">7 <td valign="bottom" align="center" style=" font-size:10pt;">1 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">8 <td valign="bottom" align="center" style=" font-size:10pt;">64 <td valign="bottom" align="center" style=" font-size:10pt;">8 <td valign="bottom" align="center" style=" font-size:10pt;">1 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">9 <td valign="bottom" align="center" style=" font-size:10pt;">81 <td valign="bottom" align="center" style=" font-size:10pt;">9 <td valign="bottom" align="center" style=" font-size:10pt;">1 </tr> <tr> <td valign="bottom" align="center" style=" font-size:10pt;">10 <td valign="bottom" align="center" style=" font-size:10pt;">100 <td valign="bottom" align="center" style=" font-size:10pt;">10 <td valign="bottom" align="center" style=" font-size:10pt;">1 </tr> </table> <p> The implementation of the <code>MultivariateMatrixFunction jacobian() for this problem looks like this (The x parameter is an ArrayList containing the independent values of the data set): </p> <source> 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); jacobian[i][1] = x.get(i); jacobian[i][2] = 1.0; } return jacobian; } public MultivariateMatrixFunction jacobian() { return new MultivariateMatrixFunction() { private static final long serialVersionUID = -8673650298627399464L; public double[][] value(double[] point) { return jacobian(point); } }; } </source> <p> Note that if for some reason the derivative of the objective function with respect to its variables is difficult to obtain, <a href="http://en.wikipedia.org/wiki/Numerical_differentiation">Numerical differentiation can be used. </p> <p> The implementation of the <code>double[] value(double[] point) method, which returns a <code>double array containing the values the objective function returns per given independent value and the current set of variables or parameters, can be seen below: </p> <source> public double[] value(double[] variables) { double[] values = new double[x.size()]; for (int i = 0; i < values.length; ++i) { values[i] = (variables[0] * x.get(i) + variables[1]) * x.get(i) + variables[2]; } return values; } </source> <p> Below is the the class containing all the implementation details (Taken from the Apache Commons Math <b>org.apache.commons.math3.optimization.general.LevenbergMarquardtOptimizerTest): </p> <source> private static class QuadraticProblem implements DifferentiableMultivariateVectorFunction, Serializable { private static final long serialVersionUID = 7072187082052755854L; private List<Double> x; private List<Double> y; public QuadraticProblem() { x = new ArrayList<Double>(); y = new ArrayList<Double>(); } public void addPoint(double x, double y) { this.x.add(x); this.y.add(y); } public double[] calculateTarget() { double[] target = new double[y.size()]; for (int i = 0; i < y.size(); i++) { target[i] = y.get(i).doubleValue(); } return target; } 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); jacobian[i][1] = x.get(i); jacobian[i][2] = 1.0; } return jacobian; } public double[] value(double[] variables) { double[] values = new double[x.size()]; for (int i = 0; i < values.length; ++i) { values[i] = (variables[0] * x.get(i) + variables[1]) * x.get(i) + variables[2]; } return values; } public MultivariateMatrixFunction jacobian() { return new MultivariateMatrixFunction() { private static final long serialVersionUID = -8673650298627399464L; public double[][] value(double[] point) { return jacobian(point); } }; } } </source> <p> The below code shows how to go about using the above class and a LevenbergMarquardtOptimizer instance to produce an optimal set of quadratic curve fitting parameters: </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(); 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); final double[] optimalValues = optimum.getPoint(); System.out.println("A: " + optimalValues[0]); System.out.println("B: " + optimalValues[1]); System.out.println("C: " + optimalValues[2]); </source> <p> If you run the above sample you will see the following printed by the console: </p> <pre> A: 7.998832172372726 B: 10.001841530162448 C: 16.324008168386605 </pre> </dd> <p> In addition to least squares solving, the <a 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/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/direct/PowellOptimizer.html"> PowellOptimizer</a> provides an optimization method for non-differentiable functions. </p> </subsection> </section> </body> </document>

Java example source code file (optimization.xml) This example Java source code file (optimization.xml) is included in the alvinalexander.com "Java Source Code Warehouse" project. The intent of this project is to help you "Learn Java by Example" TM. Learn more about this Java project at its project page. Java - Java tags/keywords apache, arraylist, differentiablemultivariatevectorfunction, jacobian, levenbergmarquardtoptimizer, license, matrix, multivariatematrixfunction, quadraticproblem, the, these, this The optimization.xml Java example source code <?xml version="1.0"?> <!-- Licensed to the Apache Software Foundation (ASF) under one or more contributor license agreements. See the NOTICE file distributed with this work for additional information regarding copyright ownership. The ASF licenses this file to You under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. --> <?xml-stylesheet type="text/xsl" href="./xdoc.xsl"?> <document url="optimization.html"> <properties> <title>The Commons Math User Guide - Optimization </properties> <body> <section name="12 Optimization"> <p>The contents of this section currently describes deprecated classes. Please refer to the new <a href="../apidocs/org/apache/commons/math3/optim/package-summary.html">API description</a>. </p> <p>Least squares optimizers are not in this package anymore, they have been moved in a dedicated least-squares sub-package described in the <a href="./leastsquares.html">least squares section. </p> <subsection name="12.1 Overview" href="overview"> <p> The optimization package provides algorithms to optimize (i.e. either minimize or maximize) some objective or cost function. The package is split in several sub-packages dedicated to different kind of functions or algorithms. <ul> <li>the univariate package handles univariate scalar functions, <li>the linear package handles multivariate vector linear functions with linear constraints,</li> <li>the direct package handles multivariate scalar functions using direct search methods (i.e. not using derivatives),</li> <li>the general package handles multivariate scalar or vector functions using derivatives.</li> <li>the fitting package handles curve fitting by univariate real functions </ul> </p> <p> The top level optimization package provides common interfaces for the optimization algorithms provided in sub-packages. The main interfaces defines defines optimizers and convergence checkers. The functions that are optimized by the algorithms provided by this package and its sub-packages are a subset of the one defined in the <code>analysis package, namely the real and vector valued functions. These functions are called objective function here. When the goal is to minimize, the functions are often called cost function, this name is not used in this package. </p> <p> The type of goal, i.e. minimization or maximization, is defined by the enumerated <a href="../apidocs/org/apache/commons/math3/optimization/GoalType.html"> GoalType</a> which has only two values: MAXIMIZE and MINIMIZE. </p> <p> Optimizers are the algorithms that will either minimize or maximize, the objective function by changing its input variables set until an optimal set is found. There are only four interfaces defining the common behavior of optimizers, one for each supported type of objective function: <ul> <li> UnivariateOptimizer</a> for <li> MultivariateOptimizer</a> for <li> DifferentiableMultivariateOptimizer</a> for <li> DifferentiableMultivariateVectorOptimizer</a> for </ul> </p> <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/MultivariateVectorFunction.html"> non-differentiable multivariate vectorial function</a> by converting it to a . </p> <p> These algorithms usage is very similar to root-finding algorithms usage explained in the analysis package. The main difference is that the <code>solve methods in root finding algorithms is replaced by <code>optimize methods. </p> </subsection> <subsection name="12.3 Linear Programming" href="linear"> <p> This package provides an implementation of George Dantzig's simplex algorithm for solving linear optimization problems with linear equality and inequality constraints. </p> </subsection> <subsection name="12.4 Direct Methods" href="direct"> <p> Direct search methods only use cost function values, they don't need derivatives and don't either try to compute approximation of the derivatives. According to a 1996 paper by Margaret H. Wright (<a href="http://cm.bell-labs.com/cm/cs/doc/96/4-02.ps.gz">Direct Search Methods: Once Scorned, Now Respectable</a>), they are used when either the computation of the derivative is impossible (noisy functions, unpredictable discontinuities) or difficult (complexity, computation cost). In the first cases, rather than an optimum, a <em>not too bad point is desired. In the latter cases, an optimum is desired but cannot be reasonably found. In all cases direct search methods can be useful. </p> <p> Simplex-based direct search methods are based on comparison of the cost function values at the vertices of a simplex (which is a set of n+1 points in dimension n) that is updated by the algorithms steps. </p> <p> The instances can be built either in single-start or in multi-start mode. Multi-start is a traditional way to try to avoid being trapped in a local minimum and miss the global minimum of a function. It can also be used to verify the convergence of an algorithm. In multi-start mode, the <code>minimizesmethod returns the best minimum found after all starts, and the <code>etMinima method can be used to retrieve all minima from all starts (including the one already provided by the <code>minimizes method). </p> <p> The <code>direct package provides four solvers: <ul> <li>the classical <li>Virginia Torczon's <li>Nikolaus Hansen's <li>Mike Powell's such that a cost function J = sum(w<sub>i(mesi - modi)2) is minimized. The various (target<sub>i - modeli(pk)) terms are called residuals. They represent the deviation between a set of target values target<sub>i and theoretical values computed from models model<sub>i depending on free parameters pk. The w<sub>i factors are weights. One classical use case is when the target values are experimental observations or measurements. </p> <p> Solving a least-squares problem is finding the free parameters p<sub>k of the theoretical models such that they are close to the target values, i.e. when the residual are small. </p> <p> Two optimizers are available in the general package, both devoted to least-squares problems. The first one is based on the <a href="../apidocs/org/apache/commons/math3/optimization/general/GaussNewtonOptimizer.html"> Gauss-Newton</a> method. The second one is the 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 method is the point from which the optimizer will start its search for the optimal point. </p> <dl> <dt>Quadratic Problem Example <dd> We are looking to find the best parameters [a, b, c] for the quadratic function <b>f(x) = a x2 + b x + c. 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> <td valign="bottom" align="center" style=" font-size:10pt;">X

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