We are given the polyhedron $X=\{x:Ax\le b,x\ge 0\}$ and the point $y\in X$. We want to find a point $x \in X$ such that $d(x,y)$ is maximized. The function $d(x,y)$ represents the distance between the two points. With regard to this problem, I have the following question:
Are you aware of some distance function $d(x,y)$ such that the problem can be formulated as a linear program or be solved very efficiently?
I was trying to use $d(x,y)=\sum_{i=1}^n|x_i-y_i|$ where $x_i$ and $y_i$ are elements of $n$-vectors $x$ and $y$, respectively. The issue is that I can only formulate the problem as a mixed-integer linear program, not a linear program. Any suggestions?