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Numerical Optimization: Difference between revisions
→Cauchy Point Algorithms
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<math> p_k^s = \operatorname{argmin}_{p \in \mathbb{R}^n} f_k + g_k^Tp </math> s.t. <math>\Vert p \Vert \leq \Delta_k </math><br> | <math> p_k^s = \operatorname{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> | and <math>\tau_k</math> minimizes our quadratic model along the line <math>p_k^s</math>:<br> | ||
<math>\tau_k = \ | <math>\tau_k = \operatorname{argmin}_{\tau \geq 0} m_k(\tau p_k^s)</math> s.t. <math>\Vert \tau p_k^s \Vert \leq \Delta_k </math><br> | ||
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 = | 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} | \begin{cases} | ||