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I know how to projects onto a linear subspace of $\mathbb R^3$, but how to project a point $x$ onto an polyhedron given as the intersection of three halfspaces $$ \langle y_1, x \rangle \ge c_1 \mbox{ and } \langle y_2, x \rangle \ge c_2 \mbox{ and } \langle y_3, x \rangle \ge c_3? $$

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As far as I know there are no closed form for the projection of a point onto an arbitrary polyhedron. You must solve

$$min_z ||x-z||_2$$

$$ \langle y_i^T z\rangle \geq c_i , i=1,\ldots$$

For some special cases there might be particularly efficient algorithms (as for instance the unitary simplex).

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