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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?

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This is an instance of linear programming, so you can solve it with any LP solver.

Any instance of linear programming can be represented in this form. For instance, minimize $c^\top w$ subject to $Aw \le b$ can be represented by setting $b^i_\max = b_i$, $x^i_j = A_{i,j}$, $b^i_\min=-\infty$, $x^0_j=c_j$. Therefore, you can't expect any special-case algorithm that is more efficient than general-purpose linear programming.

The worst-case complexity of linear programming is polynomial but a bit messy, but in practice it is often very efficient on real-world instances.

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