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Numerical Optimization: Difference between revisions
→Cauchy Point Algorithms
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and <math>\tau_k</math> minimizes our quadratic model along the line <math>p_k^s</math>:<br> | and <math>\tau_k</math> minimizes our quadratic model along the line <math>p_k^s</math>:<br> | ||
<math>\tau_k = argmin_{\tau \geq 0} m_k(\tau p_k^s)</math> s.t. <math>\Vert \tau p_k^s \leq \Delta_k </math><br> | <math>\tau_k = argmin_{\tau \geq 0} m_k(\tau p_k^s)</math> s.t. <math>\Vert \tau p_k^s \leq \Delta_k </math><br> | ||
This can be written explicitly as <math>p_k^ | This can be written explicitly as <math>p_k^c = - \tau_k \frac{\Delta_k}{\Vert g_K \Vert} g_k</math> where <math>\tau_k = | ||
\begin{cases} | |||
1 & \text{if }g_k^T B-k g_k \leq 0;\\ | |||
\min(\Vert g_k \Vert ^3/(\Delta_k g_k^T B_k g_k), 1) & \text{otherwise} | |||
\end{cases} | |||
</math> | |||
==Resources== | ==Resources== | ||
* [https://link.springer.com/book/10.1007%2F978-0-387-40065-5 Numerical Optimization by Nocedal and Wright (2006)] | * [https://link.springer.com/book/10.1007%2F978-0-387-40065-5 Numerical Optimization by Nocedal and Wright (2006)] | ||