Let the program linear :

$$\begin{cases} min C_{1}x_{1}+C_{2}x_{2}\\ A_{11}x_{1}+A_{12}x_{2}≤b_{1}\\ A_{21}x_{1}+A_{22}x_{2}=b_{2}\\ x_{1}≥0 \end{cases}$$

Where $A$ matrix $(m_{1}+m_{2})×(n_{1}+n_{2})$

And : $x,C\in\mathbb{R}^{n_{1}+n_{2}}$

And. $b\in\mathbb{R}^{m_{1}+m_{2}}$

Then find the dual program of the above primal program

I know that if we have :

$$\begin{cases} min C^{T}x\\ Ax≥b\\ x≥0 \end{cases}$$

Then the dual program is :

$$\begin{cases} max b^{T}y\\ A^{T}y≤C\\ y≥0 \end{cases}$$

But I don't know how I get the dual program in the first program !!


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