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\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. Then 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 $\delta$ neighborhood of $\mathbf{q}$. However, since $\mathbf{q} \in \mathbb{S}^{n-1}$, the quadratic approximation is performed on the tangent space $T_{\mathbf{q}} \mathbb{S}^{n-1} = \{\delta in \in  \mathbb{R}^n | \mathbf{q}^T \mathbf{\delta} = 0\}$. 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}.