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Up: 4.1 Optimization Methods
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4.1.3 Steepest-descent
The modeling of the target function can be done by a
linear approximation of
based on a Taylor-series
expansion of first order
![\begin{displaymath}
f(\vec{x}_0 + \vec{p}) \approx
f(\vec{x}_0) + ( {\mathop{\nabla }\nolimits f(\vec{x}_0)} ) ^{\cal T} \vec{p}
\end{displaymath}](img103.gif) |
(4.6) |
where
denotes a small step away from the current point
.
A reduction of the function can be achieved by a step
where
is a
negative number with a large absolute value. The direction can be
found by solving the following minimization problem
![\begin{displaymath}
\min_{\vec{p} \in \mathbb{R}^{n}}
\frac{({\mathop{\nabla }...
...ec{x}_0)})^{\cal T} \vec{p}}
{\vert\vert\vec{p}\vert\vert _2}
\end{displaymath}](img106.gif) |
(4.7) |
which gives direction
![\begin{displaymath}
\vec{p} = - {\mathop{\nabla }\nolimits f(\vec{x}_0)}
\end{displaymath}](img107.gif) |
(4.8) |
along the negative gradient of the target function f.
R. Plasun