What if objective function $Z$ is also in the constraints?

Question

What if objective function $Z$ is in the constraints? To construct the dual form for this problem? how do I approach to this problem?

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Maximize $\;\;\;\;\;\;\; z$

subject to $$\;\;\;z - \sum_{i=1}^{m}a_{ij}x_i \leq 0 $$

$$\;\;\;\;\;\;\;\;\;\;\;\; \sum_{i=1}^{m}x_i = 1$$

$$ x_i \geq 0\;\;\;\;\; \forall i=1,...,m$$

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Answer 1

This doesn't change anything (and in the general case, $z$ appears in the constraints, since otherwise maximizing $z$ would be very easy).

Here, the dual problem is minimizing $z$ subject to the constraints $z - a_{i,j}y_j \geq 0$ and $\sum y_j \geq 1$, as you can see by taking the transpose of the matrix.