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I am considering the following question:

Suppose that two constraints in a system are cx $\leq$ 1 and dx $\leq$ 1, where c and d are linearly dependent. If cd $\geq$ 0 does this mean that one of c or d is redundant?

Intuitively, I would say the answer is yes, one of the constraints is redundant. Because cd $\geq$ 0, both c and d are pointing in the same direction (i.e. they are both positive or both negative), any constraint imposed by one will either make redundant or be made redundant by the other. I am having trouble fully forming this idea and proving it more formally - any tips?

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Yes, one of them is redundant. We can assume that both c and d are not zero, because if one of them were 0 that inequality is clearly redundant.

If both c and d are not zero:

$$x \le \frac{1}{c} \qquad \text{and} \qquad x \le \frac{1}{d}$$

And clearly one of these inequalities is redundant because you only need to consider the one with the smallest bound and the other will be automatically fulfilled.

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