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Next: <a href="Nonlinear-Least_002dSquares-TRS-Double-Dogleg.html#Nonlinear-Least_002dSquares-TRS-Double-Dogleg" accesskey="n" rel="next">Nonlinear Least-Squares TRS Double Dogleg</a>, Previous: <a href="Nonlinear-Least_002dSquares-TRS-Levenberg_002dMarquardt-with-Geodesic-Acceleration.html#Nonlinear-Least_002dSquares-TRS-Levenberg_002dMarquardt-with-Geodesic-Acceleration" accesskey="p" rel="previous">Nonlinear Least-Squares TRS Levenberg-Marquardt with Geodesic Acceleration</a>, Up: <a href="Nonlinear-Least_002dSquares-TRS-Overview.html#Nonlinear-Least_002dSquares-TRS-Overview" accesskey="u" rel="up">Nonlinear Least-Squares TRS Overview</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Dogleg"></a>
<h4 class="subsection">39.2.3 Dogleg</h4>
<a name="index-Dogleg-algorithm"></a>
<a name="index-nonlinear-least-squares_002c-dogleg"></a>

<p>This is Powell&rsquo;s dogleg method, which finds an approximate
solution to the trust region subproblem, by restricting
its search to a piecewise linear &ldquo;dogleg&rdquo; path,
composed of the origin, the Cauchy point which represents
the model minimizer along the steepest descent direction,
and the Gauss-Newton point, which is the overall minimizer
of the unconstrained model. The Gauss-Newton step is calculated by
solving
</p>
<div class="example">
<pre class="example">J_k \delta_gn = -f_k
</pre></div>

<p>which is the main computational task for each iteration,
but only needs to be performed once per iteration. If
the Gauss-Newton point is inside the trust region, it is
selected as the step. If it is outside, the method then
calculates the Cauchy point, which is located along the
gradient direction. If the Cauchy point is also outside
the trust region, the method assumes that it is still far
from the minimum and so proceeds along the gradient
direction, truncating the step at the trust region
boundary. If the Cauchy point is inside the trust region,
with the Gauss-Newton point outside, the method
uses a dogleg step, which is a linear combination of the
gradient direction and the Gauss-Newton direction, stopping at the trust
region boundary.
</p>



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