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I have the following simplified optimization problem:

\begin{align}\max &\quad ax+by\\\text{s.t.}&\quad0 \le x \le \overline X\\&\quad0 \le y \le\overline Y\\&\quad z = E-x+\beta\cdot y\end{align}

where $E$, $\beta$, $a$, $b$, $\overline X$ and $\overline Y$ are parameters and the rest are variables. What would be the dual objective function for this optimization problem?

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For each variable, you need to define a constraint in the dual problem and likewise, for each constraint in the primal problem, you will have a dual variable.

\begin{align}\min&\quad\overline X\times D_1 + \overline Y \times D_2 + E\times D_3\\\text{s.t.}&\quad D_1+D_3 \ge a\\&\quad D_2-\beta D_3 \ge b\\&\quad D_3 = 0\\&\quad D_1 \ge 0\\&\quad D_2 \ge 0\end{align}

After simplifications you will have the following dual problem:

\begin{align}\min&\quad\overline X \times D_1 + \overline Y \times D_2\\\text{s.t.}&\quad D_1\ge a\\&\quad D_2\ge b\\&\quad D_1\ge 0\\&\quad D_2\ge 0\end{align}

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