So I have a question, and I'm not even sure what fields might best answer it (so even any hints or keywords to search would be helpful to me). I have a handful of small models, and I want to know which is the minimum as I change parameters. The model coefficients are non-negative:
$(1) z = 4b\\ (2) z = a + 2b\\ (3) z = 3a + b\\ (4) z = 5a$
I'm not really interested in the value of a or b, but instead their ratio $r=b/a$. The punchline:
$r < 0.5: \text{(1) is the minimum.}\\ r = 0.5: \text{(1) = (2).}\\ 0.5 < r < 2: \text{(2) is the minimum.}\\ r = 2: \text{(2) = (3) = (4).}\\ r > 2: \text{(4) is the minimum.}$
For each model I could check it against every other, find where they intersect, and then test each interval, but I'm going to have a large number of models (1,000 - 10,000) and I'd rather avoid that exponential behavior. I'll also have more than 2 parameters in the future, but for now let's not worry about that. For now, I'd settle for a way to find any point where one model is equal to one or more others.
This is similar to solving a system of linear equations, except I don't need all of them to be true for some solution, only 2 or more. I feel like linear algebra can do this, but I'm over my head trying to figure out how. I have found the following so far, perhaps it may help. Putting my models into a matrix:
$$M = \begin{bmatrix}0&4\\1&2\\3&1\\5&0\end{bmatrix}$$ I found $M\cdot\begin{bmatrix}2\\1\end{bmatrix} = \begin{bmatrix}4\\4\\7\\5\end{bmatrix}$ and $M\cdot\begin{bmatrix}1\\2\end{bmatrix} = \begin{bmatrix}8\\5\\5\\5\end{bmatrix}$. In the first, that row 1 and 2 both equal 4 indicates that model 1 = model 2, z=4 when a=2 and b=1. Similarly in the second that the last 3 rows are equal indicates that models 2=3=4, z=5 when a=1 and b=2. From this it follows that if you subtract a broadcasted row from M, any 0 in the result indicates an intersection with that row: $$(M-\begin{bmatrix}1&2\\1&2\\1&2\\1&2\end{bmatrix})\cdot \begin{bmatrix}1\\2\end{bmatrix} = \begin{bmatrix}3\\0\\0\\0\end{bmatrix}$$
That indicates that model 2 intersects models 3 and 4 at a=1 and b=2. That's almost the form to be solved by eiganvalue decomposition, but not quite. I'm not sure if any of that helps. Is there anything out there that might apply to this situation? I'd appreciate any advice.