How to code check if system of inequalities has a solution

Question

I need to find algorithm that can check if system like this

x1 * k + b > y1, x2 * k + b > y2, ..., xn * k + b < yn

has solution(s) where i substitute x[i] and y[i] and unknown varialbes are k,b.

Answer 1

If only the last inequality is "<" and all the other ones are ">". Here is how to check:

Transform the system to: b > y1 - x1 * k, b > y2 - x2 * k, ..., b < yn - xn * k

And it is easy to see that whether the original system has solutions is equivalent to the system yn - xn * k > y1 - x1 * k, yn - xn * k > y2 - x2 * k, ... has solutions

And it is equivalent to yn - y1 > (xn - x1) * k, yn - y2 > (xn - x2) * k, ... has solutions or not.

Then you need to discuss the signs of xn - xk, whether they are zero, positive or negative, and you can further transform the system to simpler form. For example, if xn - x1 > 0 and xn - x2 <0, it will look like this: k < (yn - y1)/(xn - x1), k > (yn - y2)/(xn - x2), ...

And then it is easy to check whether the new simple system has solutions or not which is equivalent to the original system has solutions or not.

Answer 2

You question is equivalent to asking if the points are linearly separable (with one class of points corresponding to the inequalities with >, the others with <).

You can use the Perceptron Algorithm to find a separating line if one exists. That wikipedia page provides some alternative algorithms too.