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<h1>Introduction <span class="docs-time d-none d-md-flex">Last update: August 23, 2021</span></h1>
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<p>
PDFO (Powell's Derivative-Free Optimization solvers) is a cross-platform package providing interfaces for using the late Professor <a class="theme-link" href="https://www.zhangzk.net/powell.html" target="_blank">M. J. D. Powell</a>'s <a class="theme-link" href="https://en.wikipedia.org/wiki/Derivative-free_optimization" target="_blank">derivative-free optimization</a> solvers, including <a class="theme-link" href="https://en.wikipedia.org/wiki/UOBYQA" target="_blank">UOBYQA</a>, <a class="theme-link" href="https://en.wikipedia.org/wiki/NEWUOA" target="_blank">NEWUOA</a>, <a class="theme-link" href="https://en.wikipedia.org/wiki/BOBYQA" target="_blank">BOBYQA</a>, <a class="theme-link" href="https://en.wikipedia.org/wiki/LINCOA" target="_blank">LINCOA</a>, and <a class="theme-link" href="https://en.wikipedia.org/wiki/COBYLA" target="_blank">COBYLA</a>, which were originally implemented in Fortran 77.
</p>
<p>
Professor Powell devised these solvers to tackle <a class="theme-link external" href="http://plato.asu.edu/sub/nlores.html#general" target="_blank">general nonlinear optimization problems</a> of continuous variables with or without constraints using only <a class="theme-link" href="http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2007_03.pdf" target="_blank">function values but not derivatives</a> of the objective function or nonlinear constraint functions.
In practice, such functions are often black boxes defined by simulations.
Consequently, the corresponding optimization problems are often categorized as <a class="theme-link" href="https://en.wikipedia.org/wiki/Derivative-free_optimization" target="_blank">black-box optimization</a> or <a class="theme-link" href="https://en.wikipedia.org/wiki/Simulation-based_optimization" target="_blank">simulation-based optimization</a>.
Problem specified by explicit formulas can probably be handled by other methods more efficiently.
See the <a class="theme-link external" href="http://plato.asu.edu/sub/nlores.html#general" target="_blank">Decision Tree for Optimization Software</a> for more information.
</p>
<p>
The current version of PDFO supports MATLAB and Python.
It relies on <a class="theme-link" href="https://www.mathworks.com/help/matlab/ref/mex.html" target="_blank">MEX</a> for MATLAB and <a class="theme-link" href="https://docs.scipy.org/doc/numpy/f2py/" target="_blank">F2PY</a> for Python to compile the Fortran solvers and wrap them into user-friendly functions.
</p>
<p>
Based on Professor Powell's Fortran code, PDFO is developed by <a class="theme-link" href="https://tomragonneau.com" target="_blank">Tom M. Ragonneau</a> and <a class="theme-link" href="https://www.zhangzk.net" target="_blank">Zaikun Zhang</a> at the <a class="theme-link" href="https://www.polyu.edu.hk/ama" target="_blank">Department of Applied Mathematics</a>, <a class="theme-link" href="https://www.polyu.edu.hk" target="blank">The Hong Kong Polytechnic University</a>.
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<h1>Download <span class="docs-time d-none d-md-flex">Last update: April 26, 2023</span></h1>
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<p>
Here is the latest release of PDFO. For earlier versions, see the <a class="theme-link" href="#releases">Releases</a> section.
</p>
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<a href="https://github.com/pdfo/pdfo/archive/refs/tags/v1.3.zip"><i class="fas fa-download mr-2"></i>Version 1.3</a>
</li>
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<p>
While MATLAB users need to download this source code package to install PDFO, Python users can install PDFO via PyPI without the source code.
See the <a class="theme-link" href="#installation">Installation</a> section for more details.
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<h1>Installation <span class="docs-time d-none d-md-flex">Last update: April 22, 2023</span></h1>
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<p>
PDFO can be installed separately for MATLAB and Python.
</p>
<section class="docs-section" id="installation-matlab">
<h2>MATLAB</h2>
<h3>Prerequisites</h3>
<p>
PDFO supports MATLAB R2014a and later releases.
To use the MATLAB version of PDFO, you need first to configure the <a class="theme-link" href="https://www.mathworks.com/help/matlab/ref/mex.html" target="_blank">MEX</a> of your MATLAB so that it can compile Fortran.
To check whether your MEX is ready, run the following code in MATLAB:
</p>
<pre class="shadow-lg rounded"><code class="displayed hljs"><span class="hljs-attr">mex</span>('-v', '-setup', 'Fortran'); <span class="hljs-attr">mex</span>('-v', <span class="hljs-attr">fullfile</span>(matlabroot, 'extern', 'examples', 'refbook', 'timestwo.F')); <span class="hljs-attr">timestwo</span>(1); <span class="hljs-attr">delete</span>('timestwo.mex*'); </code></pre>
<p>
It will attempt to set up your MEX and then compile an <a class="theme-link" href="https://www.mathworks.com/help/matlab/matlab_external/create-fortran-source-mex-file.html" target="blank">example provided by MathWorks</a> for testing MEX on Fortran.
If this completes successfully, then your MEX is ready.
Otherwise, it is not, and you may try the <a class="theme-link" href="https://github.com/equipez/setup_mex" target="_blank">setup_mex package</a> at
<pre class="shadow-lg rounded"><code class="displayed json hljs">https://github.com/equipez/setup_mex</code></pre>
It will help you to set MEX up on Windows or macOS (the setup of MEX is trivial on Linux). In case setup_mex does not work, you need to consult
a local MATLAB expert or the technical support of MathWorks about "h<a class="theme-link" href="https://www.mathworks.com/help/matlab/ref/mex.html" target="_blank">ow to set up MEX</a>", which is not part of PDFO.
</p>
<h3>Installation</h3>
<p>
Download and decompress the <a class="theme-link" href="#download">source code package</a>.
You will obtain a folder containing <code>setup.m</code>.
Place this folder at the location where you want PDFO to be installed.
In MATLAB, change the directory to this folder, and execute the following command:
</p>
<pre class="shadow-lg rounded"><code class="displayed json hljs">setup</code></pre>
<p>
If this command runs successfully, PDFO is installed.
You may execute the following command in MATLAB to verify the installation:
</p>
<pre class="shadow-lg rounded"><code class="displayed json hljs">testpdfo</code></pre>
</section>
<section class="docs-section" id="installation-python">
<h2>Python</h2>
<h3>Recommended installation</h3>
<p>
To use the Python version of PDFO on Linux, Mac, or Windows, you need <a class = "theme-link" href="https://www.python.org" target="_blank">Python</a> 3.7 or above.
</p>
<p>
We highly recommend installing PDFO via <a class="theme-link" href="https://pypi.org/project/pdfo" target="_blank">PyPI</a>.
This does not need you to download the source code.
<a class="theme-link" href="https://pip.pypa.io/en/stable/installing" target="_blank">Install pip</a> in your system.
Then execute
</p>
<pre class="shadow-lg rounded"><code class="displayed json hljs"><span class="hljs-attr">python3</span> -m pip install pdfo</code></pre>
<p>
in a command shell (e.g., the <a class="theme-link" href="https://support.apple.com/guide/terminal/mac" target="_blank">terminal</a> in Linux or Mac, or the Command Shell for Windows).
If your Python 3 launcher is not <code>python3</code>, adapt the command accordingly (it may be <code>python</code> on Windows for example).
If this command runs successfully, PDFO is installed.
You may verify the installation by
</p>
<pre class="shadow-lg rounded"><code class="displayed json hljs"><span class="hljs-attr">python3</span> -m unittest pdfo.testpdfo</code></pre>
<p>
If you are an Anaconda user, PDFO is also available through the <a class="theme-link" href="https://anaconda.org/conda-forge/pdfo" target="_blank">conda installer</a>.
However, it is not managed by us.
</p>
<h3 id="recommended-installation-python">Alternative installation (using source distribution)</h3>
<p>
Alternatively, although deeply discouraged, PDFO can be installed from the source code.
It requires you to install additional Python headers, a Fortran compiler (e.g., <a class="theme-link" href="https://gcc.gnu.org/fortran/" target="_blank">gfortran</a>), and <a class="theme-link" href="https://numpy.org/doc/stable/f2py/" target="_blank">F2PY</a> (provided by <a class="theme-link" href="https://numpy.org" target="_blank">NumPy</a>).
Download and decompress the <a class="theme-link" href="#download">source code package</a>; you will obtain a folder containing
<code>setup.py</code>; in a command shell, change your directory to this folder, and then run <code class="bkg"><span class="hljs-attr">python3</span> -m pip install ./</code> to install PDFO.
</p>
</section>
</article>
<!-- Usage -->
<article class="docs-article" id="usage">
<header class="docs-header">
<h1>Usage <span class="docs-time d-none d-md-flex">Last update: May 30, 2020</span></h1>
</header>
<section class="docs-section" id="usage-matlab">
<h2>MATLAB</h2>
<p>
PDFO provides the following MATLAB functions: <code>pdfo</code>, <code>uobyqa</code>, <code>newuoa</code>, <code>bobyqa</code>, <code>lincoa</code>, and <code>cobyla</code>.
</p>
<p>
The <code>pdfo</code> function can automatically identify the type of your problem and then call one of Powell's Fortran solvers.
The other five functions call the solver indicated by their names.
It is highly recommended to use <code>pdfo</code> instead of <code>uobyqa</code>, <code>newuoa</code>, etc.
</p>
<p>
The <code>pdfo</code> function is designed to be compatible with the <a class="theme-link" href="https://www.mathworks.com/help/optim/ug/fmincon.html" target="_blank"><code>fmincon</code></a> function available in the <a class="theme-link" href="https://www.mathworks.com/products/optimization.html" target="_blank">Optimization Toolbox</a> of MATLAB.
You can call <code>pdfo</code> in the same way as calling <a class="theme-link" href="https://www.mathworks.com/help/optim/ug/fmincon.html" target="_blank"><code>fmincon</code></a>.
In addition, <code>pdfo</code> can be called in some flexible ways that are not supported by <a class="theme-link" href="https://www.mathworks.com/help/optim/ug/fmincon.html" target="_blank"><code>fmincon</code></a>, which will be illustrated in the example below.
</p>
<p>
For the detailed syntax of these functions, use the standard <code>help</code> command of MATLAB.
For example,
</p>
<pre class="shadow-lg rounded"><code class="displayed json hljs"><span class="hljs-attr">help</span> pdfo</code></pre>
<p>
will tell you how to use the <code>pdfo</code> function.
</p>
<h3>An example</h3>
<p id="chrosen">
The following code illustrates how to minimize the chained
<a class="theme-link" href="https://en.wikipedia.org/wiki/Rosenbrock_function" target="_blank">Rosenbrock function</a>
<small>$$\sum_{i=1}^2 [(x_i - 1)^2 + 4(x_{i+1} - x_i^2)^2]$$</small>
subject to various constraints.
</p>
<div class="docs-code-block">
<script src="static/scripts/code/rosenbrock-example-matlab.js"></script>
</div>
<h3 id="options-matlab">Options</h3>
<p>
When calling <code>pdfo</code>, we may specify some options by passing a structure to <code>pdfo</code> as the last input.
Here are several useful options.
</p>
<ol>
<li>
<code>solver</code>: a string indicating which solver to use; default: <code>'uobyqa'</code> for unconstrained problems with at most 8 variables, <code>'newuoa'</code> for unconstrained problems with 9 or more variables, <code>'bobyqa'</code> for bound-constrained problems, <code>'lincoa'</code> for linearly constrained problems, and <code>'cobyla'</code> for nonlinearly constrained problems.
If you want to choose a solver, note that <a class="theme-link" href="https://en.wikipedia.org/wiki/UOBYQA" target="_blank">UOBYQA</a> and <a class="theme-link" href="https://en.wikipedia.org/wiki/NEWUOA" target="_blank">NEWUOA</a> can solve unconstrained problems, <a class="theme-link" href="https://en.wikipedia.org/wiki/NEWUOA" target="_blank">NEWUOA</a> being preferable except for rather small problems; <a class="theme-link" href="https://en.wikipedia.org/wiki/BOBYQA" target="_blank">BOBYQA</a> can solve unconstrained and bound-constrained problems; <a class="theme-link" href="https://en.wikipedia.org/wiki/LINCOA" target="_blank">LINCOA</a> can solve unconstrained, bound-constrained, and linearly constrained problems; <a class="theme-link" href="https://en.wikipedia.org/wiki/COBYLA" target="_blank">COBYLA</a> can solve general nonlinear optimization problems.
We observe that LINCOA sometimes outperforms NEWUOA on unconstrained problems. It is also worth noting that BOBYQA
evaluates the objective function only at feasible points, while LINCOA and COBYLA may explore infeasible points.
</li>
<li>
<code>maxfun</code>: maximal number of function evaluations; default: <code>500*length(x0)</code>.
</li>
<li>
<code>ftarget</code>: target function value; <code>pdfo</code> terminates once it finds a feasible point with a function value at most <code>ftarget</code>; default: <code>-inf</code>.
</li>
<li>
<code>scale</code>: a boolean value indicating whether to scale the problem according to bound constraints; if it is <code>true</code> and if all the variables have both lower and upper bounds, then the problem will be scaled so that the bound constraints become <small>\(-1 \leq x \le 1\)</small>; default: <code>false</code>.
</li>
<li>
<code>rhobeg</code>: initial trust-region radius; typically, <code>rhobeg</code> should be in the order of one tenth of the greatest expected change to a variable; <code>rhobeg</code> should be positive; default: <code>1</code> if the problem will not be scaled, and <code>0.5</code> if the problem will be scaled; in case of scaling, <code>rhobeg</code> will be used as the initial trust-region radius of the scaled problem.
</li>
<li>
<code>rhoend</code>: final trust-region radius; <code>rhoend</code> reflects the precision of the approximate solution obtained by <code>pdfo</code>; <code>rhoend</code> should be positive and not larger than <code>rhobeg</code>; default: <code>1e-6</code>; in case of scaling, <code>rhoend</code> will be used as the final trust-region radius of the scaled problem.
</li>
</ol>
<p>
For instance, to minimize the aforementioned <a class="theme-link" href="#chrosen">chained Rosenbrock function</a> without constraints by <a class="theme-link" href="https://en.wikipedia.org/wiki/LINCOA" target="_blank">LINCOA</a> with at most <small>\(50\)</small> function evaluations and a target function value <small>\(10^{-2}\)</small>, it suffices to replace <code>pdfo(@chrosen, x0)</code> in the above example by
</p>
<pre class="shadow-lg rounded"><code class="displayed hljs">pdfo(@chrosen, x0, struct('solver', 'lincoa', 'maxfun', 50, 'ftarget', 1e-2))</code></pre>
</section>
<section class="docs-section" id="usage-python">
<h2>Python</h2>
<p>
PDFO provides the following Python functions: <code>pdfo</code>, <code>uobyqa</code>, <code>newuoa</code>, <code>bobyqa</code>, <code>lincoa</code>, and <code>cobyla</code>.
</p>
<p>
The <code>pdfo</code> function can automatically identify the type of your problem and the call one of Powell's solvers.
The other five functions call the solver indicated by their names.
It is highly recommended to use <code>pdfo</code> instead of <code>uobyqa</code>, <code>newuoa</code>, etc.
</p>
<p>
The <code>pdfo</code> function is designed to be compatible with the <a class="theme-link" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html" target="_blank"><code>minimize</code></a> function available in the <code>scipy.optimize</code> module of <a class="theme-link" href="https://www.scipy.org" target="_blank">SciPy</a>.
You can call <code>pdfo</code> in exactly the same way as calling <a class="theme-link" href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize.html" target="_blank"><code>minimize</code></a> except that <code>pdfo</code> does not accept derivative arguments.
</p>
<p>
For detailed syntax of these functions, use the standard <code>help</code> command of Python.
For example,
</p>
<pre class="shadow-lg rounded"><code class="displayed json hljs"> <span class="hljs-attr">from</span> pdfo <span class="hljs-attr">import</span> pdfo; <span class="hljs-attr">help</span>(pdfo)</code></pre>
<p>
will tell you how to use <code>pdfo</code>.
</p>
<h3>An example</h3>
<p>
The following code illustrates how to minimize the <a class="theme-link" href="#chrosen">chained Rosenbrock function</a> defined above subject to various constraints.
</p>
<div class="docs-code-block">
<script src="static/scripts/code/rosenbrock-example-python.js"></script>
</div>
<h3>Method and options</h3>
<p>
When using <code>pdfo</code>, we may specify the solver by passing a string to the <a class="theme-link" href="https://en.wikipedia.org/wiki/Named_parameter" target="_blank">keyword argument</a> named <code>method</code>.
Otherwise, <code>pdfo</code> will choose the solver in the same way as <a class="theme-link" href="#options-matlab">the MATLAB version</a> does.
</p>
<p>
In addition, options may be specified by passing a dictionary to the <a class="theme-link" href="https://en.wikipedia.org/wiki/Named_parameter" target="_blank">keyword argument</a> named <code>options</code>.
Useful options include <code>maxfev</code>, <code>ftarget</code>, <code>scale</code>, <code>rhobeg</code>, and <code>rhoend</code>,
who are identical to the <a class="theme-link" href="#options-matlab">options for the MATLAB version</a> with the same names, except that the maximal number of function evaluations is named <code>maxfev</code> to align with the <code>minimize</code> function in <code>scipy.optimize</code>.
</p>
<p>
For instance, to minimize the aforementioned <a class="theme-link" href="#chrosen">chained Rosenbrock function</a> without constraints by
<a class="theme-link" href="https://en.wikipedia.org/wiki/LINCOA" target="_blank">LINCOA</a> with at most <small>\(50\)</small> function evaluations and
a target function value <small>\(10^{-2}\)</small>, it suffices to replace <code>pdfo(chrosen, x0)</code> in the above example by
</p>
<pre class="shadow-lg rounded"><code class="displayed hljs">pdfo(chrosen, x0, method='lincoa', options={'maxfev': 50, 'ftarget': 1e-2})</code></pre>
</section>
</article>
<!-- Releases -->
<article class="docs-article" id="releases">
<header class="docs-header">
<h1>Releases <span class="docs-time d-none d-md-flex">Last update: April 26, 2023</span></h1>
</header>
Here are all the releases of PDFO. For the latest version under development, check our
<a class="theme-link" href="https://github.com/pdfo/pdfo" target="_blank">repository on GitHub</a> or the mirrors on <a class="theme-link" href="https://gitlab.com/pdfo/pdfo" target="_blank">GitLab</a> and <a class="theme-link" href="https://gitee.com/pdfo/pdfo" target="_blank">Gitee</a>.
<div class="table-responsive my-4">
<table class="table table-bordered table-hover" style="width: 100%">
<colgroup>
<col span="1" style="width: 30%;">
<col span="1" style="width: 30%;">
<col span="1" style="width: 40%;">
</colgroup>
<thead>
<tr>
<th class="theme-bg-light p-3">Version number</th>
<td class="theme-bg-light p-3">Date</td>
<td class="theme-bg-light p-3">Remark</td>
</tr>
</thead>
<tbody>
<tr>
<th class="normal-weight p-3"><a class="theme-link" href="https://github.com/pdfo/pdfo/archive/refs/tags/v1.3.zip">1.3</a></th>
<td class="p-3">April 25, 2023</td>
<td class="p-3">Change PDFO's license and upgrade the compilation of the Fortran backend for Python to use <a class="theme-link" href="https://mesonbuild.com">Meson</a>.</td>
</tr>
<tr>
<th class="normal-weight p-3"><a class="theme-link" href="https://github.com/pdfo/pdfo/archive/refs/tags/v1.2.1.zip">1.2</a></th>
<td class="p-3">October 7, 2021</td>
<td class="p-3">Rebuilt of Python wheels to include <a class="theme-link" href="https://github.com/adang1345/delvewheel" target="_blank">devlewheel</a> patch for Anaconda</td>
</tr>
<tr>
<th class="normal-weight p-3"><a class="theme-link" href="https://github.com/pdfo/pdfo/releases/download/v1.1/pdfo-1.1.zip">1.1</a></th>
<td class="p-3">August 23, 2021</td>
<td class="p-3">Simplify installation for Python by providing wheels on <a class="theme-link" href="https://pypi.org/project/pdfo/#files" target="_blank">PyPI</a> for Linux, macOS, and Windows</td>
</tr>
<tr>
<th class="normal-weight p-3"><a class="theme-link" href="https://github.com/pdfo/pdfo/releases/download/v1.0/pdfo-1.0.zip">1.0</a></th>
<td class="p-3">June 10, 2020</td>
<td class="p-3">First stable version</td>
</tr>
<tr>
<th class="normal-weight p-3"><a class="theme-link" href="https://github.com/pdfo/pdfo/releases/download/v0.9/pdfo-0.9.zip">0.9</a></th>
<td class="p-3">April 19, 2020</td>
<td class="p-3">Public beta release</td>
</tr>
</tbody>
</table>
</div>
</article>
<!-- Issues -->
<article class="docs-article" id="issues">
<header class="docs-header">
<h1>Issues <span class="docs-time d-none d-md-flex">Last update: June 15, 2023</span></h1>
</header>
<p>
During the development of PDFO, some issues have occurred.
We keep here a list of the most salient ones.
</p>
<p>
In case of problems or bugs when using PDFO, you may contact us or <a class="theme-link" href="https://github.com/pdfo/pdfo/issues" target="_blank">open a new issue</a> on GitHub.
</p>
<div class="table-responsive my-4">
<table class="table table-bordered table-hover" style="width: 100%">
<colgroup>
<col span="1" style="width: 16%;">
<col span="1" style="width: 42%;">
<col span="1" style="width: 42%;">
</colgroup>
<thead>
<tr>
<th class="theme-bg-light p-3">Date</th>
<th class="theme-bg-light p-3">Description</th>
<th class="theme-bg-light p-3">Status/Solution</th>
</tr>
</thead>
<tbody>
<tr data-toggle="collapse" data-target=".collapse-issues" class="clickable">
<th class="normal-weight p-3">August 23, 2021</th>
<td class="p-3">
Python users of version 1.0 and below needed to install manually some dependencies, e.g., Python headers and a Fortran compiler.
</td>
<td class="p-3">
Starting from version 1.1, wheel distributions are available on <a class="theme-link" href="https://pypi.org/project/pdfo/#files" target="_blank">PyPI</a> for Windows, Linux, and macOS.
The wheel distributions are generated automatically using <a class="theme-link"
href="https://github.com/pdfo/pdfo/actions" target="_blank">GitHub Actions</a>. Users do not need to handle the dependencies anymore as long as
they install PDFO via <a class="theme-link" href="https://pypi.org/project/pdfo/#files" target="_blank">PyPI</a> with Python 3.7 or above.
</td>
</tr>
<tr>
<th class="normal-weight p-0">
<div class="collapse collapse-issues">
<div class="p-3">
June 15, 2020
</div>
</div>
</th>
<td class="p-0">
<div class="collapse collapse-issues">
<div class="p-3">
As of June 15, 2020, version 1.0 of PDFO could not be installed on Windows for Python 3.8 and above, because the most recent version of <a class="theme-link" href="https://software.intel.com/content/www/us/en/develop/tools/distribution-for-python.html" target="_blank">Intel Distribution for Python</a> supports only Python 3.7.
</div>
</div>
</td>
<td class="p-0">
<div class="collapse collapse-issues">
<div class="p-3">
The latest versions of Python are supported on Windows by version 1.1 and above.
</div>
</div>
</td>
</tr>
<tr>
<th class="normal-weight p-0">
<div class="collapse collapse-issues">
<div class="p-3">
April 19, 2020
</div>
</div>
</th>
<td class="p-0">
<div class="collapse collapse-issues">
<div class="p-3">
Version 0.9 does not support 64-bit Python on Windows because F2PY does not work well with MinGW-w64.
</div>
</div>
</td>
<td class="p-0">
<div class="collapse collapse-issues">
<div class="p-3">
64-bit Python is supported by version 1.0 and above on Windows.
</div>
</div>
</td>
</tr>
</tbody>
</table>
<a data-toggle="collapse" data-target=".collapse-issues" href="#">Click here to see more/less issues</a>.
</div>
</article>
<!-- References -->
<article class="docs-article" id="references">
<header class="docs-header">
<h1>Citation <span class="docs-time d-none d-md-flex">Last update: June 15, 2023</span></h1>
</header>
<section class="docs-intro">
<p>
If you use PDFO, please cite the following paper. Note that PDFO
contains improvements and bug fixes that do not exist in Powell's
original code. See Subsections 4.3–4.5 of the paper below for details.
</p>
<p>
<a class="theme-link" href="https://arxiv.org/abs/2302.13246" target="_blank">[1]</a>
T. M. Ragonneau and Z. Zhang, <a class="theme-link" href="https://arxiv.org/pdf/2302.13246.pdf" target="_blank">PDFO: a cross-platform package for
Powell's derivative-free optimization solvers</a>, <a class="theme-link" href="https://arxiv.org/abs/2302.13246" target="_blank">arXiv:2302.13246</a>, 2023.
</p>
<pre class="shadow-lg rounded"><code class="displayed hljs">@misc{Ragonneau_Zhang_2023,
title = {{PDFO}: a cross-platform package for {P}owell's derivative-free optimization solvers},
author = {Ragonneau, T. M. and Zhang, Z.},
howpublished = {arXiv:2302.13246},
year = 2023,
}</code></pre>
<p>
In addition, references for Powell's methods are as follows.
</p>
<p>
<a class="theme-link" href="https://link.springer.com/chapter/10.1007/978-94-015-8330-5_4" target="_blank">[2]</a>
M. J. D. Powell, A direct search optimization method that models the objective and constraint functions by linear interpolation, In Advances in Optimization and Numerical Analysis, eds. S. Gomez and J. P. Hennart, pages 51–67, Springer Verlag, Dordrecht, Netherlands, 1994
</p>
<p>
<a class="theme-link" href="http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2000_14.ps.gz" target="_blank">[3]</a>
M. J. D. Powell, UOBYQA: unconstrained optimization by quadratic approximation, <i>Math. Program.</i>, 92(B):555–582, 2002
</p>
<p>
<a class="theme-link" href="http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2002_08.ps.gz" target="_blank">[4]</a>
M. J. D. Powell, Least Frobenius norm updating of quadratic models that satisfy interpolation conditions. <i>Math. Program.</i>, 100:183–215, 2004
</p>
<p>
<a class="theme-link" href="http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2003_03.ps.gz" target="_blank">[5]</a>
M. J. D. Powell, On the use of quadratic models in unconstrained minimization without derivatives, <i>Optim. Methods Softw.</i>, 19:399–411, 2004
</p>
<p>
<a class="theme-link" href="http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2004_01.pdf" target="_blank">[6]</a>
M. J. D. Powell, On updating the inverse of a KKT matrix, in <i>Numerical Linear Algebra and Optimization</i>, ed. Ya-xiang Yuan, Science Press (Beijing), pp. 56–78, 2004
</p>
<p>
<a class="theme-link" href="http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2004_08.pdf" target="_blank">[7]</a>
M. J. D. Powell, The NEWUOA software for unconstrained optimization without derivatives, In <i>Large-Scale Nonlinear Optimization</i>, eds. G. Di Pillo and M. Roma, pages 255–297, Springer, New York, US, 2006
</p>
<p>
<a class="theme-link" href="http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2007_03.pdf" target="_blank">[8]</a>
M. J. D. Powell, A view of algorithms for optimization without derivatives, Technical Report DAMTP 2007/NA63, Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge, UK, 2007
</p>
<p>
<a class="theme-link" href="http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2007_05.pdf" target="_blank">[9]</a>
M. J. D. Powell, Developments of NEWUOA for minimization without derivatives, <i>IMA J. Numer. Anal.</i>, 28:649–664, 2008
</p>
<p>
<a class="theme-link" href="http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2009_06.pdf" target="_blank">[10]</a>
M. J. D. Powell, The BOBYQA algorithm for bound constrained optimization without derivatives, Technical Report DAMTP 2009/NA06, Department of Applied Mathematics and Theoretical Physics, Cambridge University, Cambridge, UK, 2009
</p>
<p>
<a class="theme-link" href="http://www.damtp.cam.ac.uk/user/na/NA_papers/NA2014_02.pdf" target="_blank">[11]</a>
M. J. D. Powell, On fast trust region methods for quadratic models with linear constraints, <i>Math. Program. Comput.</i>, 7:237–267, 2015
</p>
<p>
<a class="theme-link" href="https://tomragonneau.com/documents/thesis.pdf" target="_blank">[12]</a>
T. M. Ragonneau. <i>Model-Based Derivative-Free Optimization Methods and Software</i>, Chapter 3. PhD thesis, The Hong Kong Polytechnic University, Hong Kong SAR, China, 2022.
</p>
<p>
<a class="theme-link" href="http://www.libprima.net" target="_blank">[13]</a>
Z. Zhang, PRIMA: Reference Implementation for Powell's Methods
with Modernization and Amelioration, available at
<a class="theme-link" href="http://www.libprima.net" target="_blank">http://www.libprima.net</a>, 2023.
</p>
<h2>Remarks</h2>
<ol>
<li>
A key technique underlying the success of NEWUOA, BOBYQA, and LINCOA is the least Frobenius norm updating of quadratic models elaborated in [4] and [5].
The idea comes from the <a class="theme-link" href="https://www.jstor.org/stable/2030103?seq=1" target="_blank">least change update</a> for <a class="theme-link" href="https://epubs.siam.org/doi/abs/10.1137/1019005" target="_blank">quasi-Newton methods</a>, a vast research area initiated by the <a class="theme-link" href="https://academic.oup.com/comjnl/article/6/2/163/364776" target="_blank">DFP algorithm</a>, where P stands for Powell.
</li>
<li>
The least Frobenius norm updating is a quadratic programming problem, whose constraints correspond to the interpolation conditions.
At each iteration of Powell's algorithms, only one of the constraints is different from the previous iteration.
To solve this problem efficiently and stably, Powell designed a procedure to update the inverse of its KKT matrix along the iterations.
Such a procedure is detailed in [6], and it is indispensable for the remarkable numerical stability of NEWUOA, BOBYQA, and LINCOA.
</li>
<li>
LINCOA seeks the least value of a nonlinear function subject to linear inequality constraints without using derivatives of the objective function.
Professor Powell did not publish a paper to introduce the algorithm.
The paper [11] discusses how LINCOA solves its trust-region subproblems.
</li>
<li>
Different from PDFO, which provides interfaces for Powell's code,
<a class="theme-link" href="http://www.libprima.net" target="_blank">[13]</a> provides the modernized reference implementation for Powell's methods.
</li>
</ol>
</section>
</article>
<!-- Licence -->
<article class="docs-article" id="licence">
<header class="docs-header">
<h1>Licence <span class="docs-time d-none d-md-flex">Last update: March 11, 2023</span></h1>
</header>
<p>
PDFO is <a class="theme-link" href="https://en.wikipedia.org/wiki/Free_software" target="_blank">free software</a>.
You can redistribute it and/or modify it under the terms of the <a class="theme-link" href="https://opensource.org/license/bsd-3-clause/" target="_blank">3-Clause BSD License</a>.
</p>
<p>
PDFO is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
See the <a class="theme-link" href="https://opensource.org/license/bsd-3-clause/" target="_blank">3-Clause BSD License</a> for more details.
</p>
</article>
<!-- Acknowledgment -->
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<h1>Acknowledgment <span class="docs-time d-none d-md-flex">Last update: March 31, 2022</span></h1>
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PDFO is dedicated to the memory of the late <a class="theme-link" href="https://www.zhangzk.net/powell.html" target="_blank">Professor Powell</a> with gratitude for his inspiration and for the treasures he left to us.
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We are grateful to Professor <a class="theme-link" href="http://lsec.cc.ac.cn/~yyx/" target="_blank">Ya-xiang Yuan</a> for his everlasting encouragement and support.
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The development of PDFO is a long-term project, which would not be sustainable without the continued funds from
the <a class="theme-link" href="https://www.ugc.edu.hk/eng/rgc/" target="_blank">Hong Kong Research Grants Council</a>
(ref. PolyU 253012/17P, PolyU 153054/20P, and PolyU 153066/21P),
the <a class="theme-link" href="https://cerg1.ugc.edu.hk/hkpfs/index.html" target="_blank">Hong Kong Ph.D. Fellowship Scheme</a> (ref. PF18-24698), and
the <a class="theme-link" href="https://www.polyu.edu.hk" target="_blank">Hong Kong Polytechnic University</a> (PolyU),
in particular the <a class="theme-link" href="https://www.polyu.edu.hk/ama" target="_blank">Department of Applied Mathematics</a> (AMA).
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Dedicated to the late <a class="theme-link" href="https://www.zhangzk.net/powell.html" target="_blank">Professor M. J. D. Powell</a> FRS (1936—2015)
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<a class="theme-link" href="https://tomragonneau.com" target="_blank">Tom M. Ragonneau</a> and <a class="theme-link" href="https://www.zhangzk.net" target="_blank">Zaikun Zhang</a>
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