Removing redundant inequalities from a linear system of linear inequalities

Specific Domains Numerics
1

I have a satisfiable system of inequalities Ax ≤ b or equivalently a_i · x ≤ b_i for 1 ≤ i ≤ n. I would like to find all indexes i such that the i\text{-th} inequality is redundant.

For example in

\begin{aligned} x + y &\le 4\\ x + 2y &\le 5\\ x &\le 3\\ x &\le 4 \\ \end{aligned}

both x \le 4 and x + 2y \le 5 are redundant, the former trivially so, the latter not so much.

For my use case I only have 2 variables, so something that only applies to that dimension is fine.

But I am also interested in a general solution in term of matrix operations on A, sort of similar to a positive Gaussian elimination.

Currently I believe I have a solution, I would like to get a nicer one. It should be enough to iterate through all rows and try to maximize a_i\cdot x subject to Ax\le b and then check if for the solution c the equality a_i\cdot c\le b_i holds.