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4

If I understand your question correctly, I think you can find your answer by considering the following two primal problems. The first is \begin{alignat*}{2} & \max & x_{1}\\ & \textrm{s.t.} & x_{1}+x_{2} & \le1\\ & & x_{1} & \le1\\ & & x & \ge0 \end{alignat*} and the second is \begin{alignat*}{2} & \max &...

3

It does not look correct, and in particular the dual of an LP is an LP, so it makes no sense to have a binary variable in the dual. I suspect what led you astray was a misunderstanding of the penalty portion of the primal objective. You can rewrite the primal objective as \begin{gather*} \sum_{j=1}^{P}c_{j}x_{j}+P_{Dhd}\left[\sum_{i=1}^{F}\sum_{j=1}^{P}a_{ij}...

2

Primal Problem \begin{align} \text{minimize} \quad & \sum_{i=1}^n a_i x_i + \sum_{i=1}^n b_i z_i \\\ \text{subject to} \quad & A\mathbf x-\mathbf d \le C\mathbf z \\ & x_i \ge 0 \quad \forall i=1,\ldots,n \\ & z_i \le 0 \quad \forall i=1,\ldots,n \end{align} The dual formulation of the primal problem can be obtained by writing the ...

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