Commit faa152a4 authored by Vincent Tam's avatar Vincent Tam

Post upcoming: Math.SE unclear dual prog

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---
title: "My Dual Answer"
date: 2019-04-26T18:01:13+02:00
categories:
- math
tags:
- linear programming
- Math.SE
draft: true
---
I intended to answer [김종현's problem on Math.SE][1]. However, the programs in
the question body _aren't_ typeset in [MathJax][2]. As a result, I downvoted
and closed this question because found it _unclear_. From the proposed dual,
it seems that I _shouldn't_ interpret the primal as a linear program. Anyways,
_without_ further clarifications from OP, I _found_ no reason to look at this
further. Here's my intended answer:
> First, you have to properly write the primal as
<div> $$\begin{array}{ccc} a & b & c \\ \hline d & e & f \\ \hdashline g & h & i \end{array}$$ </div>
<div>
$$
\begin{array}{rrrrrrrrrrrrrrr}
\max \quad & z = & 3w_1 & + & 4 w_2 & + & 5w_3 & && \\
\mbox{s.t.} \quad & & w_1 & - & w_2 & & & - & \varepsilon_1 & & & & & \le 0 && \\
& & & & w_2 & - & w_3 & & & - & \varepsilon_2 & & & \le 0 && \\
& & & & & & w_3 & & & & & - & \varepsilon_3 & \le 0&& \\
& & & & & & & & 2\varepsilon_1 & + & 3\varepsilon_2 & + & 4\varepsilon_3 &\ge 1 && \\
& & w_1 & + & w_2 & + & w_3 & & & & & & & = 1, &&
\end{array}
$$
</div>
> Your claimed dual
>
> <div>
$$
\begin{array}{ll}
\max & \varepsilon_1 r_1 + \varepsilon_2 r_2 + \varepsilon_3 r_3 + (2\varepsilon_1 +3\varepsilon_2 + 4\varepsilon_3) r_4 + r_5 \\
\text{s.t. } & x_1 + x_5 \le 3 \\
& x_2 \le 4 \\
& x_3 \le 5 \\
& r_4, r_5 \text{ free}
\end{array}
$$
</div>
>
> is incorrect since it's *not* a linear program.
[1]: https://web.archive.org/web/20190426160341/https://math.stackexchange.com/questions/3203415/am-i-have-right-answer-about-dual-problem
[2]: https://www.mathjax.org/
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