I have a set $S$ of coordinates $(x,y)$, and am estimating $f(x) = ax + b$ where $a > 0$. I also happen to know that $\forall x,y((x,y) \in S \implies y < f(x))$.
The question is how I can utilize this knowledge of the upper bound on values to improve the regression result?
My intuition is to run a "normal" linear regression on all coordinates in $S$ giving $g(x)$ and then construct $g'(x) = g(x) + c$, with $c$ being the lowest number such that $\forall x,y((x,y) \in S \implies y \leq g'(x))$, e.g. such that $g'(x)$ lies as high as it can whilst still touching at least point in $S$. I do, however, have absolutely no idea if this is the best way to do it, nor how to devise an algorithm that does this efficiently.
Any help would be greatly appreciated.