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)?

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

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