The Steepest Descent Algorithm for Unconstrained Optimization and ...
The Steepest Descent Algorithm for Unconstrained Optimization and ...
The Steepest Descent Algorithm for Unconstrained Optimization and ...
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10<br />
8<br />
6<br />
4<br />
2<br />
0<br />
−2<br />
−5 0 5<br />
Figure 2: Behavior of the steepest descent algorithm <strong>for</strong> the example of<br />
Subsection 4.1.<br />
A<br />
For this problem, a<br />
=30.234, <strong>and</strong> so the linear convergence rate δ of<br />
(<br />
A/a−1 ) 2<br />
the steepest descent algorithm is bounded above by δ =<br />
A/a+1<br />
=0.8760.<br />
If we apply the steepest descent algorithm to minimize f (x) starting<br />
from x 1 =(40, −100), we obtain the iteration values shown in Table 3 <strong>and</strong><br />
shown graphically in Figure 3.<br />
4.3 Example 3: Rapid Convergence<br />
Consider the function<br />
f (x) = 1 x T Qx − c T x +10<br />
2<br />
11