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If you write the problem of finding the closest point to $x_0$ on the cone with a Lagrange multiplier, the solution must have the form $x = (\lambda A + I )^{-1}x_0$.
If you start by diagonalizing $A$, the inverse can be computed efficiently and you can search for $\lambda$ by dichotomy. The algorithm will run in $O(n^2 \log{n})$log{1/\epsilon})$2 added 2 characters in body If you write the problem of finding the closest point to$x_0$on the cone with a Lagrange multiplier, the solution must have the form$x = (\lambda A + I )^{-1}x_0$. If you start by diagonalizing$A$, the inverse can be computed efficiently and you can search for \lambda$\lambda$by dichotomy. The algorithm will run in$O(n^2 \log{n})$1 If you write the problem of finding the closest point to$x_0$on the cone with a Lagrange multiplier, the solution must have the form$x = (\lambda A + I )^{-1}x_0$. If you start by diagonalizing$A$, the inverse can be computed efficiently and you can search for \lambda by dichotomy. The algorithm will run in$O(n^2 \log{n})\$