I recently came across this variation of linear programming problem where given a set of $2$-D points, we would like to find 1 point which maximizes certain objective function. Let's say the objective function is to maximize $y = m(x+A) + B$ where $m$ is gradient, $A$ and $B$ are constants. In the example above for simplicity, assume our objective function is $y=x$.
Now the constraints for our solution are
$x > 0$, $y > 0$, $y -m(X+A) - B \ge 0 $. Essentially saying that, our solution has to be in the first quadrant, while also located in the upper part of the objective function line itself.
My observation says that the solution has to be one of the corner points (convex hull) of our set of 2-D points (input) no matter what. Empirically, I ran some simulations and the results agreed. But, is there a more formal way to verify this?