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<h3 class="section">35.1 Overview</h3>
<p>The problem of multidimensional minimization requires finding a point
x such that the scalar function,
<pre class="example"> f(x_1, ..., x_n)
</pre>
<p class="noindent">takes a value which is lower than at any neighboring point. For smooth
functions the gradient g = \nabla f vanishes at the minimum. In
general there are no bracketing methods available for the
minimization of n-dimensional functions. The algorithms
proceed from an initial guess using a search algorithm which attempts
to move in a downhill direction.
<p>Algorithms making use of the gradient of the function perform a
one-dimensional line minimisation along this direction until the lowest
point is found to a suitable tolerance. The search direction is then
updated with local information from the function and its derivatives,
and the whole process repeated until the true n-dimensional
minimum is found.
<p>The Nelder-Mead Simplex algorithm applies a different strategy. It
maintains n+1 trial parameter vectors as the vertices of a
n-dimensional simplex. In each iteration step it tries to
improve the worst vertex by a simple geometrical transformation until
the size of the simplex falls below a given tolerance.
<p>Both types of algorithms use a standard framework. The user provides a
high-level driver for the algorithms, and the library provides the
individual functions necessary for each of the steps. There are three
main phases of the iteration. The steps are,
<ul>
<li>initialize minimizer state, <var>s</var>, for algorithm <var>T</var>
<li>update <var>s</var> using the iteration <var>T</var>
<li>test <var>s</var> for convergence, and repeat iteration if necessary
</ul>
<p class="noindent">Each iteration step consists either of an improvement to the
line-minimisation in the current direction or an update to the search
direction itself. The state for the minimizers is held in a
<code>gsl_multimin_fdfminimizer</code> struct or a
<code>gsl_multimin_fminimizer</code> struct.
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