Numerical Optimization: Difference between revisions
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At each iteration, you solve a constrained optimization subproblem to find the best step <math>p</math>.<br> | At each iteration, you solve a constrained optimization subproblem to find the best step <math>p</math>.<br> | ||
<math>\min_{p \in \mathbb{R}^n} m_k(p)</math> such that <math>\Vert p \Vert < \Delta_k </math>. | <math>\min_{p \in \mathbb{R}^n} m_k(p)</math> such that <math>\Vert p \Vert < \Delta_k </math>. | ||
===Cauchy Point Algorithms=== | |||
The Cauchy point <math>p_k^c = \tau_k p_k^s</math><br> | |||
where <math>p_k^s</math> minimizes the linear model in the trust region<br> | |||
<math> p_k^s = argmin_{p \in \mathbb{R}^n} f_k + g_k^Tp </math> s.t. <math>\Vert p \Vert \leq \Delta_k </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> | |||
This can be written explicitly as <math>p_k^s = - \frac{\Delta_k}{\Vert g-K \Vert} g_k</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)] | ||