In the context of solving a convex program via projected gradient descent i am facing the following problem: $$\min_{x\in\mathbb R^2}\lVert x-y\rVert_M^2,\qquad\lVert x\rVert\le1$$ or written equivalently $$\min_{x\in\mathbb R^2}\,\langle M(x-y),x-y\rangle,\qquad\lVert x\rVert\le1$$ for a given vector $y\in\mathbb R^2$, where $M\in\mathbb R^{2\times2}$ is a positive definite matrix. While in the case that $M=\mathrm{id}$ the solution is analytical and has the form:
$$ x^*=\frac y{\max\{1,\lVert y\rVert\}} $$ Is there a closed form solution with the weighted norm variant of this problem i.e. when M is general positive definite matrix?
Mo