deletions | additions       diff --git a/paragraph_Finding_a_local_minimum__.tex b/paragraph_Finding_a_local_minimum__.tex  index d849501..1888a8e 100644  --- a/paragraph_Finding_a_local_minimum__.tex  +++ b/paragraph_Finding_a_local_minimum__.tex 
...
\paragraph{Finding a local minimum by Riemmanian trust-region method (RTM)} By the geometry above, starting from anywhere in $\Gamma$, a second-order algorithm should converge to a local minimum. Hence this is also true for \textit{the whole sphere} by symmetry.  The adopted method is a trust-region algorithm, where each iteration seeks the minimum of a local quadratic approximation in the $\delta$ $\|\mathbf{\delta}\|_2 \leq \Delta$  neighborhood of $\mathbf{q}$. However, since $\mathbf{q} \in \mathbb{S}^{n-1}$, the quadratic approximation is performed on the tangent space $T_{\mathbf{q}} \mathbb{S}^{n-1} = \{\delta \in \mathbb{R}^n | \mathbf{q}^T \mathbf{\delta} = 0\}$ and would land in a point outside the sphere. Hence, the stepwise minimum is projected back to the sphere before proceeding. And this is the \textit{Riemmanian trust-region method}. In the tangent space, the quadratic approximation takes the form of