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Next: <a href="maxima_200.html#Functions-and-Variables-for-cobyla" accesskey="n" rel="next">Functions and Variables for cobyla</a>, Previous: <a href="maxima_198.html#cobyla_002dpkg" accesskey="p" rel="previous">cobyla-pkg</a>, Up: <a href="maxima_198.html#cobyla_002dpkg" accesskey="u" rel="up">cobyla-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Introduction-to-cobyla-1"></a>
<h3 class="section">47.1 Introduction to cobyla</h3>
<p><code>fmin_cobyla</code> is a Common Lisp translation (via <code>f2cl</code>) of the
Fortran constrained optimization routine COBYLA by Powell[1][2][3].
</p>
<p>COBYLA minimizes an objective function <em>F(X)</em> subject to <em>M</em> inequality
constraints of the form
\(g(X) \ge 0\) on <em>X</em>,
where <em>X</em> is a vector of variables that has <em>N</em> components.
</p>
<p>Equality constraints <em>g(X) = 0</em> can often be implemented by a pair of inequality
constraints
\(g(X) \ge 0\) and
\(-g(X) \ge 0.\) Maxima’s interface to COBYLA
allows equality constraints and internally converts the equality
constraints to a pair of inequality constraints.
</p>
<p>The algorithm employs linear approximations to the
objective and constraint functions, the approximations being formed by
linear interpolation at <em>N+1</em> points in the space of the variables.
The interpolation points are regarded as vertices of a simplex. The
parameter <var>RHO</var> controls the size of the simplex and it is reduced
automatically from <var>RHOBEG</var> to <var>RHOEND</var>. For each <var>RHO</var> the subroutine tries
to achieve a good vector of variables for the current size, and then
<var>RHO</var> is reduced until the value <var>RHOEND</var> is reached. Therefore, <var>RHOBEG</var> and
<var>RHOEND</var> should be set to reasonable initial changes to and the required
accuracy in the variables respectively, but this accuracy should be
viewed as a subject for experimentation because it is not guaranteed.
The routine treats each constraint individually when calculating
a change to the variables, rather than lumping the constraints together
into a single penalty function. The name of the subroutine is derived
from the phrase Constrained Optimization BY Linear Approximations.
</p>
<p>References:
</p>
<p>[1] Fortran Code is from <a href="http://plato.asu.edu/sub/nlores.html#general">http://plato.asu.edu/sub/nlores.html#general</a>
</p>
<p>[2] 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 (Kluwer Academic: Dordrecht, 1994), p. 51-67.
</p>
<p>[3] M. J. D. Powell, "Direct search algorithms for optimization calculations," Acta Numerica 7, 287-336 (1998). Also available as University of Cambridge, Department of Applied Mathematics and Theoretical Physics, Numerical Analysis Group, Report NA1998/04 from <a href="http://www.damtp.cam.ac.uk/user/na/reports.html">http://www.damtp.cam.ac.uk/user/na/reports.html</a>
</p>
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Numerical-methods">Numerical methods</a>
·<a href="maxima_424.html#Category_003a-Optimization">Optimization</a>
·<a href="maxima_424.html#Category_003a-Share-packages">Share packages</a>
·<a href="maxima_424.html#Category_003a-Package-cobyla">Package cobyla</a>
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<a name="Item_003a-cobyla_002fnode_002fFunctions-and-Variables-for-cobyla"></a><hr>
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Next: <a href="maxima_200.html#Functions-and-Variables-for-cobyla" accesskey="n" rel="next">Functions and Variables for cobyla</a>, Previous: <a href="maxima_198.html#cobyla_002dpkg" accesskey="p" rel="previous">cobyla-pkg</a>, Up: <a href="maxima_198.html#cobyla_002dpkg" accesskey="u" rel="up">cobyla-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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