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I have six linear inequalities that together specify a polygon. For a given point $P$, how can I find the nearest point $P'$ that satisfies all six inequalities (if $P$ itself does not)?

inequalities

[edit: changed picture to represent regions as well as bounds]

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    On the face of it, this is a quadratic programming problem: minimize $\lVert P - P'\rVert^2$ subject to linear inequalities on $P'$. But six is a very small number of inequalities, so trying all the segments on the boundary is probably the easiest thing to do.2012-04-28

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