Problem of finding the minimum distance from a point to the complement of a convex set is a non-convex problem. Its solution set (the set of all projections) is not a singleton in general. There have been some algorithms for finding the projection onto a convex set. But in the reverse convex case, such as, projection onto the complement of an Ellipsoid, I don't know any algorithm to find a point in the set of projections.
This is a good question with lots of possible answers.
A good place to start in looking for an answer to this question is
See Section 2, starting on page 3, for the mathematical framework (the intersection of convex sets' complements are introduced in Remark 2.2). Projections onto closed convex cones are introduced in Section 1, page 3.
The boundary of a convex set and boundary of the complement of a convex set are considered relative to projections in
U. Eckhardt, W. Scherl, Z. Yu, Representation of plane curves by means of descriptors in Hough space:
http://www.math.uni-hamburg.de/home/eckhardt/HOUGH.pdf
See Chapter 9 (Reconstructon), starting on page 20.
A strange as it may seem, a good place to look in considering this question is in the projection onto coconvex (complement of a convex set) in terms of sequences of automorphisms. This is done in
K.-U. Bux, Finiteness properties of soluble arithmetic groups over global function fields, Geometry and Topology 8, 2004, 611-644:
http://msp.org/gt/2004/8-2/gt-v8-n2-p04-p.pdf
See Chapter 6 (The Moufang Property), starting on page 628.
If the complement is bonded, consider farthest points.
You will end up with a serious problem if your original set is closed, because then its complement in whatever space you use may not be, so there is a problem regarding whether there even IS a solution to that problem.
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