Consider the following optimisation problem that its size is parameterised by $n$ (width) and $m$ (depth):
Find $w_1, w_2, \ldots, w_n$ that minmises: $$ \min \Big[w_1x_1^0 + w_2x_2^0 + \ldots + w_nx_n^0 \Big] $$
Subject to:
$$ b^1_{\min} \le w_1x_1^1 + w_2x_2^1 + \ldots + w_nx_n^1 \le b^1_{\max} $$ $$ b^2_{\min} \le w_1x_1^2 + w_2x_2^2 + \ldots + w_nx_n^2 \le b^2_{\max} $$ $$ \vdots $$ $$ b^m_{\min} \le w_1x_1^m + w_2x_2^m + \ldots + w_nx_n^m \le b^m_{\max} $$
Where:
$w_1, w_2, \ldots, w_n$ are variables that we must tune (by assigning real numbers to them) in order to solve the optimisation problem above.
for any $1 \le i \le n$ and $1 \le j \le m$, $x_i^j$ is a constant real number, and $b^j_{\min}, b^j_{\max}$ are bounds.
Note that superscript $^j$ is not an exponent, but a second index. I thought it would be more readable to write $x_i^j$ than $x_{i,j}$.
Question: What's the best asymptotic worst run-time and space that we can get for a solver of this problem?
E.g. can it get as low as $O(nm)$? if not, what's the lowest we can achieve?
