Let me first make my background clear. I am a PhD student with not much knowledge in optimization but I need to do some optimization as a part of my research work. My problem is as follows:
There are two matrices P and N (both have variable rows and n columns). I have to maximize the sum of minimal Hausdorff distance between $AP$ and $N$. $A$ is a binary vector $[0,1]$ and $||A||_1$ lies between a user specified range. In short, $A$ is a vector which selects points from $P$ - let's say that point set is $p$ - such that if you add maximal Hausdorff distance(p,N) for each $p$ then it will be at its maximum. (So if you take any other point set $p$ and do the same procedure, the sum should be always less than the point set specified by $A$).
Now at the same time, I have to minimize the sum of minimal Hausdorff distances with $AP$ and $P$. In short, if a points in $P$ are in the form of a circle, then $A$ should pick the point which lies on the center of the circle.
Now, if we consider a trivial case when you want to pick only one point, then we can just exhaustively search which point satisfies the following two conditions (Also, point is always selected from P):
- Sum of distances of that point to every point in N should be maximized.
- Sum of distances of that point to every point in P should be minimized.
Is there a way to achieve this? I should mention a few more things. I am unaware of optimization limitations, for example, I have heard that its very difficult to get a binary vector, so in above example, I will get A whose values are real. That is totally fine. Just let me know if there are any other constraints and if this problem is at all possible. I can add a small example if necessary. Also, I am not looking for answer but a direction to my efforts, maybe formulate this into an equation and tell me the approach.