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Say I have the following Standard-Dual linear program:
$$max<\vec b, \vec y>$$ $$s.t.:A^Ty \le \vec c$$ $$\vec y \ge \vec 0$$ Is there a way to transform it to a Canonical-Dual and equivalent linear program, i.e., that of the following form: $$max<\vec b', \vec y'>$$ $$s.t.:A'^Ty' \le \vec c'$$ $$\vec y' \in \mathbb R^n$$

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Yes. For each variable $y_i$, just add the constraint $-y_i \leq 0$, and then you don't have to assume that $\vec{y} \geq 0$.

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